Research Article Ranks of a Constrained Hermitian Matrix...
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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 514984, 9 pages http://dx.doi.org/10.1155/2013/514984 Research Article Ranks of a Constrained Hermitian Matrix Expression with Applications Shao-Wen Yu Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China Correspondence should be addressed to Shao-Wen Yu; [email protected] Received 12 November 2012; Accepted 4 January 2013 Academic Editor: Yang Zhang Copyright © 2013 Shao-Wen Yu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression 4 − 4 ∗ 4 where is a Hermitian solution to quaternion matrix equations 1 = 1 , 1 = 2 , and 3 ∗ 3 = 3 . As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations 1 = 1 , 1 = 2 , 3 ∗ 3 = 3 , and 4 ∗ 4 = 4 , which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement 4 − 4 ∼ 3 ∗ 4 with respect to a Hermitian g-inverse ∼ 3 of 3 , which is a common solution to quaternion matrix equations 1 = 1 and 1 = 2 , are also considered. 1. Introduction roughout this paper, we denote the real number field by R, the complex number field by C, the set of all × matrices over the quaternion algebra H = { 0 + 1 + 2 + 3 | 2 = 2 = 2 = = −1, 0 , 1 , 2 , 3 ∈ R} (1) by H × , the identity matrix with the appropriate size by , the column right space, the row leſt space of a matrix over H by R(), N(), respectively, the dimension of R() by dim R(), a Hermitian g-inverse of a matrix by = ∽ which satisfies ∽ = and = ∗ , and the Moore- Penrose inverse of matrix over H by † which satisfies four Penrose equations † =, † † = † , ( † ) ∗ = † , and ( † ) ∗ = † . In this case † is unique and ( † ) ∗ = ( ∗ ) † . Moreover, and stand for the two projectors =− † , = − † induced by . Clearly, and are idempotent, Hermitian and = ∗ . By [1], for a quaternion matrix , dim R() = dim N(). dim R() is called the rank of a quaternion matrix and denoted by (). Mitra [2] investigated the system of matrix equations 1 = 1 , 1 = 2 . (2) Khatri and Mitra [3] gave necessary and sufficient con- ditions for the existence of the common Hermitian solution to (2) and presented an explicit expression for the general Hermitian solution to (2) by generalized inverses. Using the singular value decomposition (SVD), Yuan [4] investigated the general symmetric solution of (2) over the real number field R. By the SVD, Dai and Lancaster [5] considered the symmetric solution of equation ∗ = (3) over R, which was motivated and illustrated with an inverse problem of vibration theory. Groß [6], Tian and Liu [7] gave the solvability conditions for Hermitian solution and its expressions of (3) over C in terms of generalized inverses, respectively. Liu, Tian and Takane [8] investigated ranks of Hermitian and skew-Hermitian solutions to the matrix equation (3). By using the generalized SVD, Chang and Wang [9] examined the symmetric solution to the matrix equations 3 ∗ 3 = 3 , 4 ∗ 4 = 4 (4)
Transcript of Research Article Ranks of a Constrained Hermitian Matrix...
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 514984 9 pageshttpdxdoiorg1011552013514984
Research ArticleRanks of a Constrained Hermitian MatrixExpression with Applications
Shao-Wen Yu
Department of Mathematics East China University of Science and Technology Shanghai 200237 China
Correspondence should be addressed to Shao-Wen Yu yushaowenecusteducn
Received 12 November 2012 Accepted 4 January 2013
Academic Editor Yang Zhang
Copyright copy 2013 Shao-Wen YuThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression 1198624minus 1198604119883119860lowast
4where
119883 is a Hermitian solution to quaternion matrix equations 1198601119883 = 119862
1 1198831198611= 1198622 and 119860
3119883119860lowast
3= 1198623 As applications we give a new
necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations 1198601119883 = 119862
11198831198611= 1198622
1198603119883119860lowast
3= 1198623 and 119860
4119883119860lowast
4= 1198624 which was investigated by Wang and Wu 2010 by rank equalities In addition extremal ranks of
the generalized Hermitian Schur complement 1198624minus 1198604119860sim
3119860lowast
4with respect to a Hermitian g-inverse 119860sim
3of 1198603 which is a common
solution to quaternion matrix equations 1198601119883 = 119862
1and119883119861
1= 1198622 are also considered
1 Introduction
Throughout this paper we denote the real number field byRthe complex number field by C the set of all 119898 times 119899 matricesover the quaternion algebra
H = 1198860+ 1198861119894 + 1198862119895 + 1198863119896 | 1198942= 1198952
= 1198962= 119894119895119896 = minus1 119886
0 1198861 1198862 1198863isin R
(1)
by H119898times119899 the identity matrix with the appropriate size by 119868the column right space the row left space of a matrix 119860 overH by R(119860) N(119860) respectively the dimension of R(119860) bydimR(119860) a Hermitian g-inverse of a matrix 119860 by 119883 = 119860
∽
which satisfies 119860119860∽119860 = 119860 and 119883 = 119883lowast and the Moore-
Penrose inverse of matrix119860 overH by119860dagger which satisfies fourPenrose equations 119860119860dagger119860 = 119860 119860
dagger119860119860dagger= 119860dagger (119860119860
dagger)lowast=
119860119860dagger and (119860
dagger119860)lowast
= 119860dagger119860 In this case 119860dagger is unique and
(119860dagger)lowast
= (119860lowast)dagger Moreover 119877
119860and 119871
119860stand for the two
projectors 119871119860
= 119868 minus 119860dagger119860 119877119860
= 119868 minus 119860119860dagger induced by 119860
Clearly119877119860and 119871
119860are idempotent Hermitian and119877
119860= 119871119860lowast
By [1] for a quaternion matrix 119860 dimR(119860) = dimN(119860)dimR(119860) is called the rank of a quaternion matrix 119860 anddenoted by 119903(119860)
Mitra [2] investigated the system of matrix equations
1198601119883 = 119862
1 119883119861
1= 1198622 (2)
Khatri and Mitra [3] gave necessary and sufficient con-ditions for the existence of the common Hermitian solutionto (2) and presented an explicit expression for the generalHermitian solution to (2) by generalized inverses Using thesingular value decomposition (SVD) Yuan [4] investigatedthe general symmetric solution of (2) over the real numberfield R By the SVD Dai and Lancaster [5] considered thesymmetric solution of equation
119860119883119860lowast= 119862 (3)
over R which was motivated and illustrated with an inverseproblem of vibration theory Groszlig [6] Tian and Liu [7]gave the solvability conditions for Hermitian solution and itsexpressions of (3) over C in terms of generalized inversesrespectively Liu Tian and Takane [8] investigated ranksof Hermitian and skew-Hermitian solutions to the matrixequation (3) By using the generalized SVD Chang andWang[9] examined the symmetric solution to the matrix equations
1198603119883119860lowast
3= 1198623 119860
4119883119860lowast
4= 1198624 (4)
2 Journal of Applied Mathematics
over R Note that all the matrix equations mentioned aboveare special cases of
1198601119883 = 119862
1 119883119861
1= 1198622
1198603119883119860lowast
3= 1198623 119860
4119883119860lowast
4= 1198624
(5)
Wang and Wu [10] gave some necessary and sufficientconditions for the existence of the common Hermitiansolution to (5) for operators between Hilbert Clowast-modulesby generalized inverses and range inclusion of matrices Inview of the complicated computations of the generalizedinverses of matrices we naturally hope to establish a morepractical necessary and sufficient condition for system (5)over quaternion algebra to have Hermitian solution by rankequalities
As is known to us solutions tomatrix equations and ranksof solutions to matrix equations have been considered previ-ously by many authors [10ndash34] and extremal ranks of matrixexpressions can be used to characterize their rank invariancenonsingularity range inclusion and solvability conditionsof matrix equations Tian and Cheng [35] investigated themaximal and minimal ranks of 119860 minus 119861119883119862 with respect to 119883with applications Tian [36] gave the maximal and minimalranks of 119860
1minus 11986111198831198621subject to a consistent matrix equation
11986121198831198622= 1198602 Tian and Liu [7] established the solvability
conditions for (4) to have a Hermitian solution over C bythe ranks of coefficientmatricesWang and Jiang [20] derivedextreme ranks of (skew)Hermitian solutions to a quaternionmatrix equation 119860119883119860
lowast+ 119861119884119861
lowast= 119862 Wang Yu and Lin
[31] derived the extremal ranks of 1198624minus 11986041198831198614subject to a
consistent system of matrix equations
1198601119883 = 119862
1 119883119861
1= 1198622 119860
31198831198613= 1198623 (6)
over H and gave a new solvability condition to system
1198601119883 = 119862
1 119883119861
1= 1198622
11986031198831198613= 1198623 119860
41198831198614= 1198624
(7)
In matrix theory and its applications there are manymatrix expressions that have symmetric patterns or involveHermitian (skew-Hermitian) matrices For example
119860 minus 119861119883119861lowast 119860 minus 119861119883 plusmn 119883
lowast119861lowast
119860 minus 119861119883119861lowastminus 119862119884119862
lowast 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast
(8)
where 119860 = plusmn119860lowast 119861 and 119862 are given and 119883 and 119884 are
variable matrices In recent papers [7 8 37 38] Liu and Tianconsidered some maximization and minimization problemson the ranks of Hermitian matrix expressions (8)
Define a Hermitian matrix expression
119891 (119883) = 1198624minus 1198604119883119860lowast
4 (9)
where 1198624= 119862lowast
4 we have an observation that by investigating
extremal ranks of (9) where 119883 is a Hermitian solution to asystem of matrix equations
1198601119883 = 119862
1 119883119861
1= 1198622 119860
3119883119860lowast
3= 1198623 (10)
A new necessary and sufficient condition for system (5) tohave Hermitian solution can be given by rank equalitieswhich is more practical than one given by generalizedinverses and range inclusion of matrices
It is well known that Schur complement is one of themostimportant matrix expressions in matrix theory there havebeen many results in the literature on Schur complementsand their applications [39ndash41] Tian [36 42] has investigatedthe maximal and minimal ranks of Schur complements withapplications
Motivated by the workmentioned above we in this paperinvestigate the extremal ranks of the quaternion Hermitianmatrix expression (9) subject to the consistent system ofquaternion matrix equations (10) and its applications InSection 2 we derive the formulas of extremal ranks of (9)with respect to Hermitian solution of (10) As applications inSection 3 we give a new necessary and sufficient conditionfor the existence of Hermitian solution to system (5) byrank equalities In Section 4 we derive extremal ranks ofgeneralized Hermitian Schur complement subject to (2) Wealso consider the rank invariance problem in Section 5
2 Extremal Ranks of (9) Subject to System (10)Corollary 8 in [10] over Hilbert Clowast-modules can be changedinto the following lemma over H
Lemma 1 Let 1198601 1198621
isin H119898times119899 1198611 1198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 be given and 119865 = 119861
lowast
11198711198601
119872 = 119878119871119865 119878 =
11986031198711198601
119863 = 119862lowast
2minus119861lowast
1119860dagger
11198621 119869 = 119860
dagger
11198621+119865dagger119863 119866 = 119862
3minus1198603(119869+
1198711198601
119871lowast
119865119869lowast)119860lowast
3 then the following statements are equivalent
(1) the system (10) have a Hermitian solution(2) 1198623= 119862lowast
3
11986011198622= 11986211198611 119860
1119862lowast
1= 1198621119860lowast
1 119861
lowast
11198622= 119862lowast
21198611 (11)
1198771198601
1198621= 0 119877
119865119863 = 0 119877
119872119866 = 0 (12)
(3) 1198623= 119862lowast
3 the equalities in (11) hold and
119903 [1198601 1198621] = 119903 (1198601) 119903 [
11986011198621
119861lowast
1119862lowast
2
] = 119903 [1198601
119861lowast
1
]
119903[[
[
11986011198621119860lowast
3
119861lowast
1119862lowast
2119860lowast
3
1198603
1198623
]]
]
= 119903[
[
1198601
119861lowast
1
1198603
]
]
(13)
In that case the general Hermitian solution of (10) can beexpressed as
119883 = 119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
+ 1198711198601
1198711198651198711198721198811198711198651198711198601
+ 1198711198601
119871119865119881lowast1198711198721198711198651198711198601
(14)
where 119881 is Hermitian matrix over H with compatible size
Journal of Applied Mathematics 3
Lemma 2 (see Lemma 24 in [24]) Let 119860 isin H119898times119899 119861 isin
H119898times119896 119862 isin H119897times119899 119863 isin H119895times119896 and 119864 isin H119897times119894 Then the followingrank equalities hold
(a) 119903(119862119871119860) = 119903 [ 119860
119862] minus 119903(119860)
(b) 119903 [ 119861 119860119871119862 ] = 119903 [ 119861 1198600 119862
] minus 119903(119862)
(c) 119903 [ 119862119877119861119860 ] = 119903 [ 119862 0
119860 119861] minus 119903(119861)
(d) 119903 [ 119860 119861119871119863119877119864119862 0
] = 119903 [119860 119861 0
119862 0 119864
0 119863 0
] minus 119903(119863) minus 119903(119864)
Lemma 2 plays an important role in simplifying ranks ofvarious block matrices
Liu and Tian [38] has given the following lemma over afield The result can be generalized to H
Lemma 3 Let 119860 = plusmn119860lowastisin H119898times119898 119861 isin H119898times119899 and 119862 isin H119901times119898
be given then
max119883isinH119899times119901
119903 [119860 minus 119861119883119862 ∓ (119861119883119862)lowast]
= min119903 [119860 119861 119862lowast] 119903 [
119860 119861
119861lowast0] 119903 [
119860 119862lowast
119862 0]
min119883isinH119899times119901
119903 [119860 minus 119861119883119862 ∓ (119861119883119862)lowast]
= 2119903 [119860 119861 119862lowast] +max 119904
1 1199042
(15)
where
1199041= 119903 [
119860 119861
119861lowast0] minus 2119903 [
119860 119861 119862lowast
119861lowast0 0
]
1199042= 119903 [
119860 119862lowast
119862 0] minus 2119903 [
119860 119861 119862lowast
119862 0 0]
(16)
IfR(119861) sube R(119862lowast)
max119883
119903 [119860 minus 119861119883119862 minus (119861119883119862)lowast] = min119903 [119860 119862
lowast] 119903 [
119860 119861
119861lowast0]
max119883
119903 [119860 minus 119861119883119862 minus (119861119883119862)lowast] = min119903 [119860 119862
lowast] 119903 [
119860 119861
119861lowast0]
(17)
Now we consider the extremal ranks of the matrixexpression (9) subject to the consistent system (10)
Theorem 4 Let 1198601 1198621 1198611 1198622 1198603 and 119862
3be defined as
Lemma 11198624isin H119905times119905 and 119860
4isin H119905times119899Then the extremal ranks
of the quaternionmatrix expression119891(119883) defined as (9) subjectto system (10) are the following
max 119903 [119891 (119883)] = min 119886 119887 (18)
where
119886 = 119903
[[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
(19)
min 119903 [119891 (119883)] = 2119903
[[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]]
]
+ 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903
[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
(20)
Proof By Lemma 1 the general Hermitian solution of thesystem (10) can be expressed as
119883 = 119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
+ 1198711198601
1198711198651198711198721198811198711198651198711198601
+ 1198711198601
119871119865119881lowast1198711198721198711198651198711198601
(21)
where 119881 is Hermitian matrix over H with appropriate sizeSubstituting (21) into (9) yields
119891 (119883) = 1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast
+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4
minus 11986041198711198601
1198711198651198711198721198811198711198651198711198601
119860lowast
4
minus 11986041198711198601
119871119865119881lowast1198711198721198711198651198711198601
119860lowast
4
(22)
4 Journal of Applied Mathematics
Put
1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4= 119860
119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
= 1198691015840
11986041198711198601
119871119865119871119872= 119873
1198711198651198711198601
119860lowast
4= 119875
(23)
then
119891 (119883) = 119860 minus 119873119881119875 minus (119873119881119875)lowast (24)
Note that119860 = 119860lowast andR(119873) sube R(119875
lowast) Thus applying (17) to
(24) we get the following
max 119903 [119891 (119883)] = max119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= min119903 [119860 119875lowast] 119903 [
119860 119873
119873lowast
0]
min 119903 [119891 (119883)] = min119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= 2119903 [119860 119875lowast] + 119903 [
119860 119873
119873lowast
0] minus 2119903 [
119860 119873
119875 0]
(25)
Now we simplify the ranks of block matrices in (25)In view of Lemma 2 block Gaussian elimination (11)
(12) and (23) we have the following
119903 (119865) = 119903 (119861lowast
11198711198601
) = 119903 [119861lowast
1
1198601
] minus 119903 (1198601)
119903 (119872) = 119903 (119878119871119865) = 119903 [
119878
119865] minus 119903 (119865)
= 119903 [
11986031198711198601
119861lowast
11198711198601
] minus 119903 (119865)
= 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 (1198601) minus 119903 (119865)
119903 [119860 119875lowast] = 119903 [1198624 minus 1198604119869119860
lowast
4119875lowast]
= 119903 [1198624minus 1198604119869119860lowast
411986041198711198601
0 119865] minus 119903 (119865)
= 119903[
[
1198624minus 1198604119869119860lowast
41198604
0 119861lowast
1
0 1198601
]
]
minus 119903 (119865) minus 119903 (1198601)
= 119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
minus 119903 [119861lowast
1
1198601
]
119903 [119860 119873
119873lowast
0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119872lowast119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[
[
1198624minus 11986041198691015840119860lowast
41198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
0 1198603
0 0 0
0 119861lowast
10 0 0
0 1198601
0 0 0
]]]]]]
]
minus 2119903 (119872) minus 2119903 (119865) minus 2119903 (1198601)
= 119903
[[[[[[[[[[[[
[
1198624
1198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
11986031198691015840119860lowast
41198603
0 0 0
119861lowast
11198691015840119860lowast
4119861lowast
10 0 0
11986011198691015840119860lowast
41198601
0 0 0
]]]]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
11986241198604
0 0 0
119860lowast
40 119860
lowast
31198611
119860lowast
1
0 1198603
minus1198623
minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
119903 [119860 119873
119875 0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[[[[[
[
11986241198604
0 0
119860lowast
40 119861
1119860lowast
1
0 1198603minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 [119861lowast
1
1198601
]
Journal of Applied Mathematics 5
= 119903
[[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[[
[
1198603
119861lowast
1
1198601
]]
]
minus 119903 [119861lowast
1
1198601
]
(26)
Substituting (26) into (25) yields (18) and (20)
In Theorem 4 letting 1198624vanish and 119860
4be 119868 with
appropriate size respectively we have the following
Corollary 5 Assume that 1198601 1198621
isin H119898times119899 11986111198622
isin
H119899times119904 1198603isin H119903times119899 and 119862
3isin H119903times119903 are given then the maximal
and minimal ranks of the Hermitian solution 119883 to the system(10) can be expressed as
max 119903 (119883) = min 119886 119887 (27)
where
119886 = 119899 + 119903 [119862lowast
2
1198621
] minus 119903 [119861lowast
1
1198601
]
119887 = 2119899 + 119903
[[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
min 119903 (119883) = 2119903 [119862lowast
2
1198621
]
+ 119903[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]
]
minus 2119903
[[[[
[
119860311986221198603119862lowast
1
119862lowast
21198611119862lowast
2119860lowast
1
119862111986111198621119860lowast
1
]]]]
]
(28)
InTheorem 4 assuming that 1198601 1198611 1198621 and 119862
2vanish
we have the following
Corollary 6 Suppose that the matrix equation 1198603119883119860lowast
3= 1198623
is consistent then the extremal ranks of the quaternion matrixexpression 119891(119883) defined as (9) subject to 119860
3119883119860lowast
3= 1198623are the
following
max 119903 [119891 (119883)]
= min
119903 [1198624 1198604] 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903 (1198603)
min 119903 [119891 (119883)] = 2119903 [1198624 1198604]
+ 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903[
[
0 119860lowast
4
11986041198624
1198603
0
]
]
(29)
3 A Practical Solvability Condition forHermitian Solution to System (5)
In this section we use Theorem 4 to give a necessary andsufficient condition for the existence of Hermitian solutionto system (5) by rank equalities
Theorem 7 Let 1198601 1198621
isin H119898times119899 11986111198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 119860
4isin H119905times119899 and 119862
4isin H119905times119905be given then
the system (5) have Hermitian solution if and only if 1198623= 119862lowast
3
(11) (13) hold and the following equalities are all satisfied
119903 [1198604 1198624] = 119903 (1198604) (30)
119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
= 119903[
[
1198604
119861lowast
1
1198601
]
]
(31)
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 2119903
[[[[
[
1198604
1198603
119861lowast
1
1198601
]]]]
]
(32)
Proof It is obvious that the system (5) have Hermitiansolution if and only if the system (10) haveHermitian solutionand
min 119903 [119891 (119883)] = 0 (33)
where 119891(119883) is defined as (9) subject to system (10) Let1198830be
aHermitian solution to the system (5) then1198830is aHermitian
solution to system (10) and1198830satisfies119860
41198830119860lowast
4= 1198624 Hence
Lemma 1 yields 1198623
= 119862lowast
3 (11) (13) and (30) It follows
6 Journal of Applied Mathematics
from
[[[[[[
[
119868 0 0 0 0
0 119868 0 0 0
119860311988300 119868 0 0
119861lowast
111988300 0 119868 0
119860111988300 0 0 119868
]]]]]]
]
times
[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
times
[[[[[
[
119868 minus1198830119860lowast
40 0 0
0 119868 0 0 0
0 0 119868 0 0
0 0 0 119868 0
0 0 0 0 119868
]]]]]
]
=
[[[[[
[
0 119860lowast
4119860lowast
31198611119860lowast
1
1198604
0 0 0 0
1198603
0 0 0 0
119861lowast
10 0 0 0
1198601
0 0 0 0
]]]]]
]
(34)
that (32) holds Similarly we can obtain (31)Conversely assume that 119862
3= 119862lowast
3 (11) (13) hold then by
Lemma 1 system (10) have Hermitian solution By (20) (31)-(32) and
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
ge 119903[[[
[
1198604
1198603
119861lowast
1
1198601
]]]
]
+ 119903[
[
1198604
119861lowast
1
1198601
]
]
(35)
we can get
min 119903 [119891 (119883)] le 0 (36)
However
min 119903 [119891 (119883)] ge 0 (37)
Hence (33) holds implying that the system (5) have Hermi-tian solution
ByTheorem 7 we can also get the following
Corollary 8 Suppose that 1198603 1198623 1198604 and 119862
4are those in
Theorem 7 then the quaternion matrix equations 1198603119883119860lowast
3=
1198623and 119860
4119883119860lowast
4= 1198624have common Hermitian solution if and
only if (30) hold and the following equalities are satisfied
119903 [1198603 1198623] = 119903 (1198603)
119903 [
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
= 2119903 [1198603
1198604
]
(38)
Corollary 9 Suppose that11986011198621isin H119898times119899119861
11198622isin H119899times119904 and
119860 119861 isin H119899times119899 are Hermitian Then 119860 and 119861 have a common
Hermitian g-inverse which is a solution to the system (2) if andonly if (11) holds and the following equalities are all satisfied
119903[[
[
11986011198621119860
119861lowast
1119862lowast
2119860
119860 119860
]]
]
= 119903[
[
1198601
119861lowast
1
119860
]
]
119903[[
[
11986011198621119861
119861lowast
1119862lowast
2119861
119861 119861
]]
]
= 119903[
[
1198601
119861lowast
1
119861
]
]
(39)
119903
[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861 119861 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]
]
= 2119903
[[[[
[
119861
119860
119861lowast
1
1198601
]]]]
]
(40)
4 Extremal Ranks of Schur ComplementSubject to (2)
As is well known for a given block matrix
119872 = [119860 119861
119861lowast119863] (41)
where 119860 and 119863 are Hermitian quaternion matrices withappropriate sizes then the Hermitian Schur complement of119860 in119872 is defined as
119878119860= 119863 minus 119861
lowast119860sim119861 (42)
where 119860sim is a Hermitian g-inverse of 119860 that is 119860sim isin 119883 |
119860119883119860 = 119860119883 = 119883lowast
Now we use Theorem 4 to establish the extremal ranksof 119878119860given by (42) with respect to 119860sim which is a solution to
system (2)
Theorem 10 Suppose 1198601 1198621isin H119898times119899 119861
1 1198622isin H119899times119904 119863 isin
H119905times119905 119861 isin H119899times119905 and 119860 isin H119899times119899 are given and system (2)is consistent then the extreme ranks of 119878
119860given by (42) with
respect to 119860sim which is a solution of (2) are the following
max1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = min 119886 119887
(43)
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
over R Note that all the matrix equations mentioned aboveare special cases of
1198601119883 = 119862
1 119883119861
1= 1198622
1198603119883119860lowast
3= 1198623 119860
4119883119860lowast
4= 1198624
(5)
Wang and Wu [10] gave some necessary and sufficientconditions for the existence of the common Hermitiansolution to (5) for operators between Hilbert Clowast-modulesby generalized inverses and range inclusion of matrices Inview of the complicated computations of the generalizedinverses of matrices we naturally hope to establish a morepractical necessary and sufficient condition for system (5)over quaternion algebra to have Hermitian solution by rankequalities
As is known to us solutions tomatrix equations and ranksof solutions to matrix equations have been considered previ-ously by many authors [10ndash34] and extremal ranks of matrixexpressions can be used to characterize their rank invariancenonsingularity range inclusion and solvability conditionsof matrix equations Tian and Cheng [35] investigated themaximal and minimal ranks of 119860 minus 119861119883119862 with respect to 119883with applications Tian [36] gave the maximal and minimalranks of 119860
1minus 11986111198831198621subject to a consistent matrix equation
11986121198831198622= 1198602 Tian and Liu [7] established the solvability
conditions for (4) to have a Hermitian solution over C bythe ranks of coefficientmatricesWang and Jiang [20] derivedextreme ranks of (skew)Hermitian solutions to a quaternionmatrix equation 119860119883119860
lowast+ 119861119884119861
lowast= 119862 Wang Yu and Lin
[31] derived the extremal ranks of 1198624minus 11986041198831198614subject to a
consistent system of matrix equations
1198601119883 = 119862
1 119883119861
1= 1198622 119860
31198831198613= 1198623 (6)
over H and gave a new solvability condition to system
1198601119883 = 119862
1 119883119861
1= 1198622
11986031198831198613= 1198623 119860
41198831198614= 1198624
(7)
In matrix theory and its applications there are manymatrix expressions that have symmetric patterns or involveHermitian (skew-Hermitian) matrices For example
119860 minus 119861119883119861lowast 119860 minus 119861119883 plusmn 119883
lowast119861lowast
119860 minus 119861119883119861lowastminus 119862119884119862
lowast 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast
(8)
where 119860 = plusmn119860lowast 119861 and 119862 are given and 119883 and 119884 are
variable matrices In recent papers [7 8 37 38] Liu and Tianconsidered some maximization and minimization problemson the ranks of Hermitian matrix expressions (8)
Define a Hermitian matrix expression
119891 (119883) = 1198624minus 1198604119883119860lowast
4 (9)
where 1198624= 119862lowast
4 we have an observation that by investigating
extremal ranks of (9) where 119883 is a Hermitian solution to asystem of matrix equations
1198601119883 = 119862
1 119883119861
1= 1198622 119860
3119883119860lowast
3= 1198623 (10)
A new necessary and sufficient condition for system (5) tohave Hermitian solution can be given by rank equalitieswhich is more practical than one given by generalizedinverses and range inclusion of matrices
It is well known that Schur complement is one of themostimportant matrix expressions in matrix theory there havebeen many results in the literature on Schur complementsand their applications [39ndash41] Tian [36 42] has investigatedthe maximal and minimal ranks of Schur complements withapplications
Motivated by the workmentioned above we in this paperinvestigate the extremal ranks of the quaternion Hermitianmatrix expression (9) subject to the consistent system ofquaternion matrix equations (10) and its applications InSection 2 we derive the formulas of extremal ranks of (9)with respect to Hermitian solution of (10) As applications inSection 3 we give a new necessary and sufficient conditionfor the existence of Hermitian solution to system (5) byrank equalities In Section 4 we derive extremal ranks ofgeneralized Hermitian Schur complement subject to (2) Wealso consider the rank invariance problem in Section 5
2 Extremal Ranks of (9) Subject to System (10)Corollary 8 in [10] over Hilbert Clowast-modules can be changedinto the following lemma over H
Lemma 1 Let 1198601 1198621
isin H119898times119899 1198611 1198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 be given and 119865 = 119861
lowast
11198711198601
119872 = 119878119871119865 119878 =
11986031198711198601
119863 = 119862lowast
2minus119861lowast
1119860dagger
11198621 119869 = 119860
dagger
11198621+119865dagger119863 119866 = 119862
3minus1198603(119869+
1198711198601
119871lowast
119865119869lowast)119860lowast
3 then the following statements are equivalent
(1) the system (10) have a Hermitian solution(2) 1198623= 119862lowast
3
11986011198622= 11986211198611 119860
1119862lowast
1= 1198621119860lowast
1 119861
lowast
11198622= 119862lowast
21198611 (11)
1198771198601
1198621= 0 119877
119865119863 = 0 119877
119872119866 = 0 (12)
(3) 1198623= 119862lowast
3 the equalities in (11) hold and
119903 [1198601 1198621] = 119903 (1198601) 119903 [
11986011198621
119861lowast
1119862lowast
2
] = 119903 [1198601
119861lowast
1
]
119903[[
[
11986011198621119860lowast
3
119861lowast
1119862lowast
2119860lowast
3
1198603
1198623
]]
]
= 119903[
[
1198601
119861lowast
1
1198603
]
]
(13)
In that case the general Hermitian solution of (10) can beexpressed as
119883 = 119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
+ 1198711198601
1198711198651198711198721198811198711198651198711198601
+ 1198711198601
119871119865119881lowast1198711198721198711198651198711198601
(14)
where 119881 is Hermitian matrix over H with compatible size
Journal of Applied Mathematics 3
Lemma 2 (see Lemma 24 in [24]) Let 119860 isin H119898times119899 119861 isin
H119898times119896 119862 isin H119897times119899 119863 isin H119895times119896 and 119864 isin H119897times119894 Then the followingrank equalities hold
(a) 119903(119862119871119860) = 119903 [ 119860
119862] minus 119903(119860)
(b) 119903 [ 119861 119860119871119862 ] = 119903 [ 119861 1198600 119862
] minus 119903(119862)
(c) 119903 [ 119862119877119861119860 ] = 119903 [ 119862 0
119860 119861] minus 119903(119861)
(d) 119903 [ 119860 119861119871119863119877119864119862 0
] = 119903 [119860 119861 0
119862 0 119864
0 119863 0
] minus 119903(119863) minus 119903(119864)
Lemma 2 plays an important role in simplifying ranks ofvarious block matrices
Liu and Tian [38] has given the following lemma over afield The result can be generalized to H
Lemma 3 Let 119860 = plusmn119860lowastisin H119898times119898 119861 isin H119898times119899 and 119862 isin H119901times119898
be given then
max119883isinH119899times119901
119903 [119860 minus 119861119883119862 ∓ (119861119883119862)lowast]
= min119903 [119860 119861 119862lowast] 119903 [
119860 119861
119861lowast0] 119903 [
119860 119862lowast
119862 0]
min119883isinH119899times119901
119903 [119860 minus 119861119883119862 ∓ (119861119883119862)lowast]
= 2119903 [119860 119861 119862lowast] +max 119904
1 1199042
(15)
where
1199041= 119903 [
119860 119861
119861lowast0] minus 2119903 [
119860 119861 119862lowast
119861lowast0 0
]
1199042= 119903 [
119860 119862lowast
119862 0] minus 2119903 [
119860 119861 119862lowast
119862 0 0]
(16)
IfR(119861) sube R(119862lowast)
max119883
119903 [119860 minus 119861119883119862 minus (119861119883119862)lowast] = min119903 [119860 119862
lowast] 119903 [
119860 119861
119861lowast0]
max119883
119903 [119860 minus 119861119883119862 minus (119861119883119862)lowast] = min119903 [119860 119862
lowast] 119903 [
119860 119861
119861lowast0]
(17)
Now we consider the extremal ranks of the matrixexpression (9) subject to the consistent system (10)
Theorem 4 Let 1198601 1198621 1198611 1198622 1198603 and 119862
3be defined as
Lemma 11198624isin H119905times119905 and 119860
4isin H119905times119899Then the extremal ranks
of the quaternionmatrix expression119891(119883) defined as (9) subjectto system (10) are the following
max 119903 [119891 (119883)] = min 119886 119887 (18)
where
119886 = 119903
[[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
(19)
min 119903 [119891 (119883)] = 2119903
[[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]]
]
+ 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903
[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
(20)
Proof By Lemma 1 the general Hermitian solution of thesystem (10) can be expressed as
119883 = 119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
+ 1198711198601
1198711198651198711198721198811198711198651198711198601
+ 1198711198601
119871119865119881lowast1198711198721198711198651198711198601
(21)
where 119881 is Hermitian matrix over H with appropriate sizeSubstituting (21) into (9) yields
119891 (119883) = 1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast
+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4
minus 11986041198711198601
1198711198651198711198721198811198711198651198711198601
119860lowast
4
minus 11986041198711198601
119871119865119881lowast1198711198721198711198651198711198601
119860lowast
4
(22)
4 Journal of Applied Mathematics
Put
1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4= 119860
119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
= 1198691015840
11986041198711198601
119871119865119871119872= 119873
1198711198651198711198601
119860lowast
4= 119875
(23)
then
119891 (119883) = 119860 minus 119873119881119875 minus (119873119881119875)lowast (24)
Note that119860 = 119860lowast andR(119873) sube R(119875
lowast) Thus applying (17) to
(24) we get the following
max 119903 [119891 (119883)] = max119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= min119903 [119860 119875lowast] 119903 [
119860 119873
119873lowast
0]
min 119903 [119891 (119883)] = min119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= 2119903 [119860 119875lowast] + 119903 [
119860 119873
119873lowast
0] minus 2119903 [
119860 119873
119875 0]
(25)
Now we simplify the ranks of block matrices in (25)In view of Lemma 2 block Gaussian elimination (11)
(12) and (23) we have the following
119903 (119865) = 119903 (119861lowast
11198711198601
) = 119903 [119861lowast
1
1198601
] minus 119903 (1198601)
119903 (119872) = 119903 (119878119871119865) = 119903 [
119878
119865] minus 119903 (119865)
= 119903 [
11986031198711198601
119861lowast
11198711198601
] minus 119903 (119865)
= 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 (1198601) minus 119903 (119865)
119903 [119860 119875lowast] = 119903 [1198624 minus 1198604119869119860
lowast
4119875lowast]
= 119903 [1198624minus 1198604119869119860lowast
411986041198711198601
0 119865] minus 119903 (119865)
= 119903[
[
1198624minus 1198604119869119860lowast
41198604
0 119861lowast
1
0 1198601
]
]
minus 119903 (119865) minus 119903 (1198601)
= 119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
minus 119903 [119861lowast
1
1198601
]
119903 [119860 119873
119873lowast
0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119872lowast119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[
[
1198624minus 11986041198691015840119860lowast
41198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
0 1198603
0 0 0
0 119861lowast
10 0 0
0 1198601
0 0 0
]]]]]]
]
minus 2119903 (119872) minus 2119903 (119865) minus 2119903 (1198601)
= 119903
[[[[[[[[[[[[
[
1198624
1198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
11986031198691015840119860lowast
41198603
0 0 0
119861lowast
11198691015840119860lowast
4119861lowast
10 0 0
11986011198691015840119860lowast
41198601
0 0 0
]]]]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
11986241198604
0 0 0
119860lowast
40 119860
lowast
31198611
119860lowast
1
0 1198603
minus1198623
minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
119903 [119860 119873
119875 0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[[[[[
[
11986241198604
0 0
119860lowast
40 119861
1119860lowast
1
0 1198603minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 [119861lowast
1
1198601
]
Journal of Applied Mathematics 5
= 119903
[[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[[
[
1198603
119861lowast
1
1198601
]]
]
minus 119903 [119861lowast
1
1198601
]
(26)
Substituting (26) into (25) yields (18) and (20)
In Theorem 4 letting 1198624vanish and 119860
4be 119868 with
appropriate size respectively we have the following
Corollary 5 Assume that 1198601 1198621
isin H119898times119899 11986111198622
isin
H119899times119904 1198603isin H119903times119899 and 119862
3isin H119903times119903 are given then the maximal
and minimal ranks of the Hermitian solution 119883 to the system(10) can be expressed as
max 119903 (119883) = min 119886 119887 (27)
where
119886 = 119899 + 119903 [119862lowast
2
1198621
] minus 119903 [119861lowast
1
1198601
]
119887 = 2119899 + 119903
[[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
min 119903 (119883) = 2119903 [119862lowast
2
1198621
]
+ 119903[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]
]
minus 2119903
[[[[
[
119860311986221198603119862lowast
1
119862lowast
21198611119862lowast
2119860lowast
1
119862111986111198621119860lowast
1
]]]]
]
(28)
InTheorem 4 assuming that 1198601 1198611 1198621 and 119862
2vanish
we have the following
Corollary 6 Suppose that the matrix equation 1198603119883119860lowast
3= 1198623
is consistent then the extremal ranks of the quaternion matrixexpression 119891(119883) defined as (9) subject to 119860
3119883119860lowast
3= 1198623are the
following
max 119903 [119891 (119883)]
= min
119903 [1198624 1198604] 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903 (1198603)
min 119903 [119891 (119883)] = 2119903 [1198624 1198604]
+ 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903[
[
0 119860lowast
4
11986041198624
1198603
0
]
]
(29)
3 A Practical Solvability Condition forHermitian Solution to System (5)
In this section we use Theorem 4 to give a necessary andsufficient condition for the existence of Hermitian solutionto system (5) by rank equalities
Theorem 7 Let 1198601 1198621
isin H119898times119899 11986111198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 119860
4isin H119905times119899 and 119862
4isin H119905times119905be given then
the system (5) have Hermitian solution if and only if 1198623= 119862lowast
3
(11) (13) hold and the following equalities are all satisfied
119903 [1198604 1198624] = 119903 (1198604) (30)
119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
= 119903[
[
1198604
119861lowast
1
1198601
]
]
(31)
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 2119903
[[[[
[
1198604
1198603
119861lowast
1
1198601
]]]]
]
(32)
Proof It is obvious that the system (5) have Hermitiansolution if and only if the system (10) haveHermitian solutionand
min 119903 [119891 (119883)] = 0 (33)
where 119891(119883) is defined as (9) subject to system (10) Let1198830be
aHermitian solution to the system (5) then1198830is aHermitian
solution to system (10) and1198830satisfies119860
41198830119860lowast
4= 1198624 Hence
Lemma 1 yields 1198623
= 119862lowast
3 (11) (13) and (30) It follows
6 Journal of Applied Mathematics
from
[[[[[[
[
119868 0 0 0 0
0 119868 0 0 0
119860311988300 119868 0 0
119861lowast
111988300 0 119868 0
119860111988300 0 0 119868
]]]]]]
]
times
[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
times
[[[[[
[
119868 minus1198830119860lowast
40 0 0
0 119868 0 0 0
0 0 119868 0 0
0 0 0 119868 0
0 0 0 0 119868
]]]]]
]
=
[[[[[
[
0 119860lowast
4119860lowast
31198611119860lowast
1
1198604
0 0 0 0
1198603
0 0 0 0
119861lowast
10 0 0 0
1198601
0 0 0 0
]]]]]
]
(34)
that (32) holds Similarly we can obtain (31)Conversely assume that 119862
3= 119862lowast
3 (11) (13) hold then by
Lemma 1 system (10) have Hermitian solution By (20) (31)-(32) and
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
ge 119903[[[
[
1198604
1198603
119861lowast
1
1198601
]]]
]
+ 119903[
[
1198604
119861lowast
1
1198601
]
]
(35)
we can get
min 119903 [119891 (119883)] le 0 (36)
However
min 119903 [119891 (119883)] ge 0 (37)
Hence (33) holds implying that the system (5) have Hermi-tian solution
ByTheorem 7 we can also get the following
Corollary 8 Suppose that 1198603 1198623 1198604 and 119862
4are those in
Theorem 7 then the quaternion matrix equations 1198603119883119860lowast
3=
1198623and 119860
4119883119860lowast
4= 1198624have common Hermitian solution if and
only if (30) hold and the following equalities are satisfied
119903 [1198603 1198623] = 119903 (1198603)
119903 [
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
= 2119903 [1198603
1198604
]
(38)
Corollary 9 Suppose that11986011198621isin H119898times119899119861
11198622isin H119899times119904 and
119860 119861 isin H119899times119899 are Hermitian Then 119860 and 119861 have a common
Hermitian g-inverse which is a solution to the system (2) if andonly if (11) holds and the following equalities are all satisfied
119903[[
[
11986011198621119860
119861lowast
1119862lowast
2119860
119860 119860
]]
]
= 119903[
[
1198601
119861lowast
1
119860
]
]
119903[[
[
11986011198621119861
119861lowast
1119862lowast
2119861
119861 119861
]]
]
= 119903[
[
1198601
119861lowast
1
119861
]
]
(39)
119903
[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861 119861 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]
]
= 2119903
[[[[
[
119861
119860
119861lowast
1
1198601
]]]]
]
(40)
4 Extremal Ranks of Schur ComplementSubject to (2)
As is well known for a given block matrix
119872 = [119860 119861
119861lowast119863] (41)
where 119860 and 119863 are Hermitian quaternion matrices withappropriate sizes then the Hermitian Schur complement of119860 in119872 is defined as
119878119860= 119863 minus 119861
lowast119860sim119861 (42)
where 119860sim is a Hermitian g-inverse of 119860 that is 119860sim isin 119883 |
119860119883119860 = 119860119883 = 119883lowast
Now we use Theorem 4 to establish the extremal ranksof 119878119860given by (42) with respect to 119860sim which is a solution to
system (2)
Theorem 10 Suppose 1198601 1198621isin H119898times119899 119861
1 1198622isin H119899times119904 119863 isin
H119905times119905 119861 isin H119899times119905 and 119860 isin H119899times119899 are given and system (2)is consistent then the extreme ranks of 119878
119860given by (42) with
respect to 119860sim which is a solution of (2) are the following
max1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = min 119886 119887
(43)
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Lemma 2 (see Lemma 24 in [24]) Let 119860 isin H119898times119899 119861 isin
H119898times119896 119862 isin H119897times119899 119863 isin H119895times119896 and 119864 isin H119897times119894 Then the followingrank equalities hold
(a) 119903(119862119871119860) = 119903 [ 119860
119862] minus 119903(119860)
(b) 119903 [ 119861 119860119871119862 ] = 119903 [ 119861 1198600 119862
] minus 119903(119862)
(c) 119903 [ 119862119877119861119860 ] = 119903 [ 119862 0
119860 119861] minus 119903(119861)
(d) 119903 [ 119860 119861119871119863119877119864119862 0
] = 119903 [119860 119861 0
119862 0 119864
0 119863 0
] minus 119903(119863) minus 119903(119864)
Lemma 2 plays an important role in simplifying ranks ofvarious block matrices
Liu and Tian [38] has given the following lemma over afield The result can be generalized to H
Lemma 3 Let 119860 = plusmn119860lowastisin H119898times119898 119861 isin H119898times119899 and 119862 isin H119901times119898
be given then
max119883isinH119899times119901
119903 [119860 minus 119861119883119862 ∓ (119861119883119862)lowast]
= min119903 [119860 119861 119862lowast] 119903 [
119860 119861
119861lowast0] 119903 [
119860 119862lowast
119862 0]
min119883isinH119899times119901
119903 [119860 minus 119861119883119862 ∓ (119861119883119862)lowast]
= 2119903 [119860 119861 119862lowast] +max 119904
1 1199042
(15)
where
1199041= 119903 [
119860 119861
119861lowast0] minus 2119903 [
119860 119861 119862lowast
119861lowast0 0
]
1199042= 119903 [
119860 119862lowast
119862 0] minus 2119903 [
119860 119861 119862lowast
119862 0 0]
(16)
IfR(119861) sube R(119862lowast)
max119883
119903 [119860 minus 119861119883119862 minus (119861119883119862)lowast] = min119903 [119860 119862
lowast] 119903 [
119860 119861
119861lowast0]
max119883
119903 [119860 minus 119861119883119862 minus (119861119883119862)lowast] = min119903 [119860 119862
lowast] 119903 [
119860 119861
119861lowast0]
(17)
Now we consider the extremal ranks of the matrixexpression (9) subject to the consistent system (10)
Theorem 4 Let 1198601 1198621 1198611 1198622 1198603 and 119862
3be defined as
Lemma 11198624isin H119905times119905 and 119860
4isin H119905times119899Then the extremal ranks
of the quaternionmatrix expression119891(119883) defined as (9) subjectto system (10) are the following
max 119903 [119891 (119883)] = min 119886 119887 (18)
where
119886 = 119903
[[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
(19)
min 119903 [119891 (119883)] = 2119903
[[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]]
]
+ 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903
[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
(20)
Proof By Lemma 1 the general Hermitian solution of thesystem (10) can be expressed as
119883 = 119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
+ 1198711198601
1198711198651198711198721198811198711198651198711198601
+ 1198711198601
119871119865119881lowast1198711198721198711198651198711198601
(21)
where 119881 is Hermitian matrix over H with appropriate sizeSubstituting (21) into (9) yields
119891 (119883) = 1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast
+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4
minus 11986041198711198601
1198711198651198711198721198811198711198651198711198601
119860lowast
4
minus 11986041198711198601
119871119865119881lowast1198711198721198711198651198711198601
119860lowast
4
(22)
4 Journal of Applied Mathematics
Put
1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4= 119860
119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
= 1198691015840
11986041198711198601
119871119865119871119872= 119873
1198711198651198711198601
119860lowast
4= 119875
(23)
then
119891 (119883) = 119860 minus 119873119881119875 minus (119873119881119875)lowast (24)
Note that119860 = 119860lowast andR(119873) sube R(119875
lowast) Thus applying (17) to
(24) we get the following
max 119903 [119891 (119883)] = max119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= min119903 [119860 119875lowast] 119903 [
119860 119873
119873lowast
0]
min 119903 [119891 (119883)] = min119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= 2119903 [119860 119875lowast] + 119903 [
119860 119873
119873lowast
0] minus 2119903 [
119860 119873
119875 0]
(25)
Now we simplify the ranks of block matrices in (25)In view of Lemma 2 block Gaussian elimination (11)
(12) and (23) we have the following
119903 (119865) = 119903 (119861lowast
11198711198601
) = 119903 [119861lowast
1
1198601
] minus 119903 (1198601)
119903 (119872) = 119903 (119878119871119865) = 119903 [
119878
119865] minus 119903 (119865)
= 119903 [
11986031198711198601
119861lowast
11198711198601
] minus 119903 (119865)
= 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 (1198601) minus 119903 (119865)
119903 [119860 119875lowast] = 119903 [1198624 minus 1198604119869119860
lowast
4119875lowast]
= 119903 [1198624minus 1198604119869119860lowast
411986041198711198601
0 119865] minus 119903 (119865)
= 119903[
[
1198624minus 1198604119869119860lowast
41198604
0 119861lowast
1
0 1198601
]
]
minus 119903 (119865) minus 119903 (1198601)
= 119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
minus 119903 [119861lowast
1
1198601
]
119903 [119860 119873
119873lowast
0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119872lowast119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[
[
1198624minus 11986041198691015840119860lowast
41198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
0 1198603
0 0 0
0 119861lowast
10 0 0
0 1198601
0 0 0
]]]]]]
]
minus 2119903 (119872) minus 2119903 (119865) minus 2119903 (1198601)
= 119903
[[[[[[[[[[[[
[
1198624
1198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
11986031198691015840119860lowast
41198603
0 0 0
119861lowast
11198691015840119860lowast
4119861lowast
10 0 0
11986011198691015840119860lowast
41198601
0 0 0
]]]]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
11986241198604
0 0 0
119860lowast
40 119860
lowast
31198611
119860lowast
1
0 1198603
minus1198623
minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
119903 [119860 119873
119875 0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[[[[[
[
11986241198604
0 0
119860lowast
40 119861
1119860lowast
1
0 1198603minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 [119861lowast
1
1198601
]
Journal of Applied Mathematics 5
= 119903
[[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[[
[
1198603
119861lowast
1
1198601
]]
]
minus 119903 [119861lowast
1
1198601
]
(26)
Substituting (26) into (25) yields (18) and (20)
In Theorem 4 letting 1198624vanish and 119860
4be 119868 with
appropriate size respectively we have the following
Corollary 5 Assume that 1198601 1198621
isin H119898times119899 11986111198622
isin
H119899times119904 1198603isin H119903times119899 and 119862
3isin H119903times119903 are given then the maximal
and minimal ranks of the Hermitian solution 119883 to the system(10) can be expressed as
max 119903 (119883) = min 119886 119887 (27)
where
119886 = 119899 + 119903 [119862lowast
2
1198621
] minus 119903 [119861lowast
1
1198601
]
119887 = 2119899 + 119903
[[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
min 119903 (119883) = 2119903 [119862lowast
2
1198621
]
+ 119903[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]
]
minus 2119903
[[[[
[
119860311986221198603119862lowast
1
119862lowast
21198611119862lowast
2119860lowast
1
119862111986111198621119860lowast
1
]]]]
]
(28)
InTheorem 4 assuming that 1198601 1198611 1198621 and 119862
2vanish
we have the following
Corollary 6 Suppose that the matrix equation 1198603119883119860lowast
3= 1198623
is consistent then the extremal ranks of the quaternion matrixexpression 119891(119883) defined as (9) subject to 119860
3119883119860lowast
3= 1198623are the
following
max 119903 [119891 (119883)]
= min
119903 [1198624 1198604] 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903 (1198603)
min 119903 [119891 (119883)] = 2119903 [1198624 1198604]
+ 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903[
[
0 119860lowast
4
11986041198624
1198603
0
]
]
(29)
3 A Practical Solvability Condition forHermitian Solution to System (5)
In this section we use Theorem 4 to give a necessary andsufficient condition for the existence of Hermitian solutionto system (5) by rank equalities
Theorem 7 Let 1198601 1198621
isin H119898times119899 11986111198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 119860
4isin H119905times119899 and 119862
4isin H119905times119905be given then
the system (5) have Hermitian solution if and only if 1198623= 119862lowast
3
(11) (13) hold and the following equalities are all satisfied
119903 [1198604 1198624] = 119903 (1198604) (30)
119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
= 119903[
[
1198604
119861lowast
1
1198601
]
]
(31)
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 2119903
[[[[
[
1198604
1198603
119861lowast
1
1198601
]]]]
]
(32)
Proof It is obvious that the system (5) have Hermitiansolution if and only if the system (10) haveHermitian solutionand
min 119903 [119891 (119883)] = 0 (33)
where 119891(119883) is defined as (9) subject to system (10) Let1198830be
aHermitian solution to the system (5) then1198830is aHermitian
solution to system (10) and1198830satisfies119860
41198830119860lowast
4= 1198624 Hence
Lemma 1 yields 1198623
= 119862lowast
3 (11) (13) and (30) It follows
6 Journal of Applied Mathematics
from
[[[[[[
[
119868 0 0 0 0
0 119868 0 0 0
119860311988300 119868 0 0
119861lowast
111988300 0 119868 0
119860111988300 0 0 119868
]]]]]]
]
times
[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
times
[[[[[
[
119868 minus1198830119860lowast
40 0 0
0 119868 0 0 0
0 0 119868 0 0
0 0 0 119868 0
0 0 0 0 119868
]]]]]
]
=
[[[[[
[
0 119860lowast
4119860lowast
31198611119860lowast
1
1198604
0 0 0 0
1198603
0 0 0 0
119861lowast
10 0 0 0
1198601
0 0 0 0
]]]]]
]
(34)
that (32) holds Similarly we can obtain (31)Conversely assume that 119862
3= 119862lowast
3 (11) (13) hold then by
Lemma 1 system (10) have Hermitian solution By (20) (31)-(32) and
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
ge 119903[[[
[
1198604
1198603
119861lowast
1
1198601
]]]
]
+ 119903[
[
1198604
119861lowast
1
1198601
]
]
(35)
we can get
min 119903 [119891 (119883)] le 0 (36)
However
min 119903 [119891 (119883)] ge 0 (37)
Hence (33) holds implying that the system (5) have Hermi-tian solution
ByTheorem 7 we can also get the following
Corollary 8 Suppose that 1198603 1198623 1198604 and 119862
4are those in
Theorem 7 then the quaternion matrix equations 1198603119883119860lowast
3=
1198623and 119860
4119883119860lowast
4= 1198624have common Hermitian solution if and
only if (30) hold and the following equalities are satisfied
119903 [1198603 1198623] = 119903 (1198603)
119903 [
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
= 2119903 [1198603
1198604
]
(38)
Corollary 9 Suppose that11986011198621isin H119898times119899119861
11198622isin H119899times119904 and
119860 119861 isin H119899times119899 are Hermitian Then 119860 and 119861 have a common
Hermitian g-inverse which is a solution to the system (2) if andonly if (11) holds and the following equalities are all satisfied
119903[[
[
11986011198621119860
119861lowast
1119862lowast
2119860
119860 119860
]]
]
= 119903[
[
1198601
119861lowast
1
119860
]
]
119903[[
[
11986011198621119861
119861lowast
1119862lowast
2119861
119861 119861
]]
]
= 119903[
[
1198601
119861lowast
1
119861
]
]
(39)
119903
[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861 119861 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]
]
= 2119903
[[[[
[
119861
119860
119861lowast
1
1198601
]]]]
]
(40)
4 Extremal Ranks of Schur ComplementSubject to (2)
As is well known for a given block matrix
119872 = [119860 119861
119861lowast119863] (41)
where 119860 and 119863 are Hermitian quaternion matrices withappropriate sizes then the Hermitian Schur complement of119860 in119872 is defined as
119878119860= 119863 minus 119861
lowast119860sim119861 (42)
where 119860sim is a Hermitian g-inverse of 119860 that is 119860sim isin 119883 |
119860119883119860 = 119860119883 = 119883lowast
Now we use Theorem 4 to establish the extremal ranksof 119878119860given by (42) with respect to 119860sim which is a solution to
system (2)
Theorem 10 Suppose 1198601 1198621isin H119898times119899 119861
1 1198622isin H119899times119904 119863 isin
H119905times119905 119861 isin H119899times119905 and 119860 isin H119899times119899 are given and system (2)is consistent then the extreme ranks of 119878
119860given by (42) with
respect to 119860sim which is a solution of (2) are the following
max1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = min 119886 119887
(43)
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Put
1198624minus 1198604(119869 + 119871
1198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
)119860lowast
4= 119860
119869 + 1198711198601
119871119865119869lowast+ 1198711198601
119871119865119872dagger119866(119872dagger)lowast
1198711198651198711198601
= 1198691015840
11986041198711198601
119871119865119871119872= 119873
1198711198651198711198601
119860lowast
4= 119875
(23)
then
119891 (119883) = 119860 minus 119873119881119875 minus (119873119881119875)lowast (24)
Note that119860 = 119860lowast andR(119873) sube R(119875
lowast) Thus applying (17) to
(24) we get the following
max 119903 [119891 (119883)] = max119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= min119903 [119860 119875lowast] 119903 [
119860 119873
119873lowast
0]
min 119903 [119891 (119883)] = min119881
119903 (119860 minus 119873119881119875 minus (119873119881119875)lowast)
= 2119903 [119860 119875lowast] + 119903 [
119860 119873
119873lowast
0] minus 2119903 [
119860 119873
119875 0]
(25)
Now we simplify the ranks of block matrices in (25)In view of Lemma 2 block Gaussian elimination (11)
(12) and (23) we have the following
119903 (119865) = 119903 (119861lowast
11198711198601
) = 119903 [119861lowast
1
1198601
] minus 119903 (1198601)
119903 (119872) = 119903 (119878119871119865) = 119903 [
119878
119865] minus 119903 (119865)
= 119903 [
11986031198711198601
119861lowast
11198711198601
] minus 119903 (119865)
= 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 (1198601) minus 119903 (119865)
119903 [119860 119875lowast] = 119903 [1198624 minus 1198604119869119860
lowast
4119875lowast]
= 119903 [1198624minus 1198604119869119860lowast
411986041198711198601
0 119865] minus 119903 (119865)
= 119903[
[
1198624minus 1198604119869119860lowast
41198604
0 119861lowast
1
0 1198601
]
]
minus 119903 (119865) minus 119903 (1198601)
= 119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
minus 119903 [119861lowast
1
1198601
]
119903 [119860 119873
119873lowast
0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119872lowast119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[
[
1198624minus 11986041198691015840119860lowast
41198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
0 1198603
0 0 0
0 119861lowast
10 0 0
0 1198601
0 0 0
]]]]]]
]
minus 2119903 (119872) minus 2119903 (119865) minus 2119903 (1198601)
= 119903
[[[[[[[[[[[[
[
1198624
1198604
0 0 0
119860lowast
40 119860lowast
31198611119860lowast
1
11986031198691015840119860lowast
41198603
0 0 0
119861lowast
11198691015840119860lowast
4119861lowast
10 0 0
11986011198691015840119860lowast
41198601
0 0 0
]]]]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
11986241198604
0 0 0
119860lowast
40 119860
lowast
31198611
119860lowast
1
0 1198603
minus1198623
minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
= 119903
[[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
119903 [119860 119873
119875 0] = 119903 [
1198624minus 11986041198691015840119860lowast
411986041198711198601
119871119865119871119872
119877119865lowast119877119860lowast
1
119860lowast
40
]
= 119903
[[[[[[[[[[
[
11986241198604
0 0
119860lowast
40 119861
1119860lowast
1
0 1198603minus11986031198622minus1198603119862lowast
1
0 119861lowast
1minus119862lowast
21198611minus119862lowast
2119860lowast
1
0 1198601minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[
[
1198603
119861lowast
1
1198601
]
]
minus 119903 [119861lowast
1
1198601
]
Journal of Applied Mathematics 5
= 119903
[[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[[
[
1198603
119861lowast
1
1198601
]]
]
minus 119903 [119861lowast
1
1198601
]
(26)
Substituting (26) into (25) yields (18) and (20)
In Theorem 4 letting 1198624vanish and 119860
4be 119868 with
appropriate size respectively we have the following
Corollary 5 Assume that 1198601 1198621
isin H119898times119899 11986111198622
isin
H119899times119904 1198603isin H119903times119899 and 119862
3isin H119903times119903 are given then the maximal
and minimal ranks of the Hermitian solution 119883 to the system(10) can be expressed as
max 119903 (119883) = min 119886 119887 (27)
where
119886 = 119899 + 119903 [119862lowast
2
1198621
] minus 119903 [119861lowast
1
1198601
]
119887 = 2119899 + 119903
[[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
min 119903 (119883) = 2119903 [119862lowast
2
1198621
]
+ 119903[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]
]
minus 2119903
[[[[
[
119860311986221198603119862lowast
1
119862lowast
21198611119862lowast
2119860lowast
1
119862111986111198621119860lowast
1
]]]]
]
(28)
InTheorem 4 assuming that 1198601 1198611 1198621 and 119862
2vanish
we have the following
Corollary 6 Suppose that the matrix equation 1198603119883119860lowast
3= 1198623
is consistent then the extremal ranks of the quaternion matrixexpression 119891(119883) defined as (9) subject to 119860
3119883119860lowast
3= 1198623are the
following
max 119903 [119891 (119883)]
= min
119903 [1198624 1198604] 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903 (1198603)
min 119903 [119891 (119883)] = 2119903 [1198624 1198604]
+ 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903[
[
0 119860lowast
4
11986041198624
1198603
0
]
]
(29)
3 A Practical Solvability Condition forHermitian Solution to System (5)
In this section we use Theorem 4 to give a necessary andsufficient condition for the existence of Hermitian solutionto system (5) by rank equalities
Theorem 7 Let 1198601 1198621
isin H119898times119899 11986111198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 119860
4isin H119905times119899 and 119862
4isin H119905times119905be given then
the system (5) have Hermitian solution if and only if 1198623= 119862lowast
3
(11) (13) hold and the following equalities are all satisfied
119903 [1198604 1198624] = 119903 (1198604) (30)
119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
= 119903[
[
1198604
119861lowast
1
1198601
]
]
(31)
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 2119903
[[[[
[
1198604
1198603
119861lowast
1
1198601
]]]]
]
(32)
Proof It is obvious that the system (5) have Hermitiansolution if and only if the system (10) haveHermitian solutionand
min 119903 [119891 (119883)] = 0 (33)
where 119891(119883) is defined as (9) subject to system (10) Let1198830be
aHermitian solution to the system (5) then1198830is aHermitian
solution to system (10) and1198830satisfies119860
41198830119860lowast
4= 1198624 Hence
Lemma 1 yields 1198623
= 119862lowast
3 (11) (13) and (30) It follows
6 Journal of Applied Mathematics
from
[[[[[[
[
119868 0 0 0 0
0 119868 0 0 0
119860311988300 119868 0 0
119861lowast
111988300 0 119868 0
119860111988300 0 0 119868
]]]]]]
]
times
[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
times
[[[[[
[
119868 minus1198830119860lowast
40 0 0
0 119868 0 0 0
0 0 119868 0 0
0 0 0 119868 0
0 0 0 0 119868
]]]]]
]
=
[[[[[
[
0 119860lowast
4119860lowast
31198611119860lowast
1
1198604
0 0 0 0
1198603
0 0 0 0
119861lowast
10 0 0 0
1198601
0 0 0 0
]]]]]
]
(34)
that (32) holds Similarly we can obtain (31)Conversely assume that 119862
3= 119862lowast
3 (11) (13) hold then by
Lemma 1 system (10) have Hermitian solution By (20) (31)-(32) and
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
ge 119903[[[
[
1198604
1198603
119861lowast
1
1198601
]]]
]
+ 119903[
[
1198604
119861lowast
1
1198601
]
]
(35)
we can get
min 119903 [119891 (119883)] le 0 (36)
However
min 119903 [119891 (119883)] ge 0 (37)
Hence (33) holds implying that the system (5) have Hermi-tian solution
ByTheorem 7 we can also get the following
Corollary 8 Suppose that 1198603 1198623 1198604 and 119862
4are those in
Theorem 7 then the quaternion matrix equations 1198603119883119860lowast
3=
1198623and 119860
4119883119860lowast
4= 1198624have common Hermitian solution if and
only if (30) hold and the following equalities are satisfied
119903 [1198603 1198623] = 119903 (1198603)
119903 [
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
= 2119903 [1198603
1198604
]
(38)
Corollary 9 Suppose that11986011198621isin H119898times119899119861
11198622isin H119899times119904 and
119860 119861 isin H119899times119899 are Hermitian Then 119860 and 119861 have a common
Hermitian g-inverse which is a solution to the system (2) if andonly if (11) holds and the following equalities are all satisfied
119903[[
[
11986011198621119860
119861lowast
1119862lowast
2119860
119860 119860
]]
]
= 119903[
[
1198601
119861lowast
1
119860
]
]
119903[[
[
11986011198621119861
119861lowast
1119862lowast
2119861
119861 119861
]]
]
= 119903[
[
1198601
119861lowast
1
119861
]
]
(39)
119903
[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861 119861 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]
]
= 2119903
[[[[
[
119861
119860
119861lowast
1
1198601
]]]]
]
(40)
4 Extremal Ranks of Schur ComplementSubject to (2)
As is well known for a given block matrix
119872 = [119860 119861
119861lowast119863] (41)
where 119860 and 119863 are Hermitian quaternion matrices withappropriate sizes then the Hermitian Schur complement of119860 in119872 is defined as
119878119860= 119863 minus 119861
lowast119860sim119861 (42)
where 119860sim is a Hermitian g-inverse of 119860 that is 119860sim isin 119883 |
119860119883119860 = 119860119883 = 119883lowast
Now we use Theorem 4 to establish the extremal ranksof 119878119860given by (42) with respect to 119860sim which is a solution to
system (2)
Theorem 10 Suppose 1198601 1198621isin H119898times119899 119861
1 1198622isin H119899times119904 119863 isin
H119905times119905 119861 isin H119899times119905 and 119860 isin H119899times119899 are given and system (2)is consistent then the extreme ranks of 119878
119860given by (42) with
respect to 119860sim which is a solution of (2) are the following
max1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = min 119886 119887
(43)
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
= 119903
[[[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]]]
]
minus 119903[[
[
1198603
119861lowast
1
1198601
]]
]
minus 119903 [119861lowast
1
1198601
]
(26)
Substituting (26) into (25) yields (18) and (20)
In Theorem 4 letting 1198624vanish and 119860
4be 119868 with
appropriate size respectively we have the following
Corollary 5 Assume that 1198601 1198621
isin H119898times119899 11986111198622
isin
H119899times119904 1198603isin H119903times119899 and 119862
3isin H119903times119903 are given then the maximal
and minimal ranks of the Hermitian solution 119883 to the system(10) can be expressed as
max 119903 (119883) = min 119886 119887 (27)
where
119886 = 119899 + 119903 [119862lowast
2
1198621
] minus 119903 [119861lowast
1
1198601
]
119887 = 2119899 + 119903
[[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]]
]
minus 2119903[
[
1198603
119861lowast
1
1198601
]
]
min 119903 (119883) = 2119903 [119862lowast
2
1198621
]
+ 119903[[[
[
1198623
119860311986221198603119862lowast
1
119862lowast
2119860lowast
3119862lowast
21198611119862lowast
2119860lowast
1
1198621119860lowast
3119862111986111198621119860lowast
1
]]]
]
minus 2119903
[[[[
[
119860311986221198603119862lowast
1
119862lowast
21198611119862lowast
2119860lowast
1
119862111986111198621119860lowast
1
]]]]
]
(28)
InTheorem 4 assuming that 1198601 1198611 1198621 and 119862
2vanish
we have the following
Corollary 6 Suppose that the matrix equation 1198603119883119860lowast
3= 1198623
is consistent then the extremal ranks of the quaternion matrixexpression 119891(119883) defined as (9) subject to 119860
3119883119860lowast
3= 1198623are the
following
max 119903 [119891 (119883)]
= min
119903 [1198624 1198604] 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903 (1198603)
min 119903 [119891 (119883)] = 2119903 [1198624 1198604]
+ 119903[
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
minus 2119903[
[
0 119860lowast
4
11986041198624
1198603
0
]
]
(29)
3 A Practical Solvability Condition forHermitian Solution to System (5)
In this section we use Theorem 4 to give a necessary andsufficient condition for the existence of Hermitian solutionto system (5) by rank equalities
Theorem 7 Let 1198601 1198621
isin H119898times119899 11986111198622
isin H119899times119904 1198603
isin
H119903times119899 1198623isin H119903times119903 119860
4isin H119905times119899 and 119862
4isin H119905times119905be given then
the system (5) have Hermitian solution if and only if 1198623= 119862lowast
3
(11) (13) hold and the following equalities are all satisfied
119903 [1198604 1198624] = 119903 (1198604) (30)
119903[[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]]
]
= 119903[
[
1198604
119861lowast
1
1198601
]
]
(31)
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 2119903
[[[[
[
1198604
1198603
119861lowast
1
1198601
]]]]
]
(32)
Proof It is obvious that the system (5) have Hermitiansolution if and only if the system (10) haveHermitian solutionand
min 119903 [119891 (119883)] = 0 (33)
where 119891(119883) is defined as (9) subject to system (10) Let1198830be
aHermitian solution to the system (5) then1198830is aHermitian
solution to system (10) and1198830satisfies119860
41198830119860lowast
4= 1198624 Hence
Lemma 1 yields 1198623
= 119862lowast
3 (11) (13) and (30) It follows
6 Journal of Applied Mathematics
from
[[[[[[
[
119868 0 0 0 0
0 119868 0 0 0
119860311988300 119868 0 0
119861lowast
111988300 0 119868 0
119860111988300 0 0 119868
]]]]]]
]
times
[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
times
[[[[[
[
119868 minus1198830119860lowast
40 0 0
0 119868 0 0 0
0 0 119868 0 0
0 0 0 119868 0
0 0 0 0 119868
]]]]]
]
=
[[[[[
[
0 119860lowast
4119860lowast
31198611119860lowast
1
1198604
0 0 0 0
1198603
0 0 0 0
119861lowast
10 0 0 0
1198601
0 0 0 0
]]]]]
]
(34)
that (32) holds Similarly we can obtain (31)Conversely assume that 119862
3= 119862lowast
3 (11) (13) hold then by
Lemma 1 system (10) have Hermitian solution By (20) (31)-(32) and
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
ge 119903[[[
[
1198604
1198603
119861lowast
1
1198601
]]]
]
+ 119903[
[
1198604
119861lowast
1
1198601
]
]
(35)
we can get
min 119903 [119891 (119883)] le 0 (36)
However
min 119903 [119891 (119883)] ge 0 (37)
Hence (33) holds implying that the system (5) have Hermi-tian solution
ByTheorem 7 we can also get the following
Corollary 8 Suppose that 1198603 1198623 1198604 and 119862
4are those in
Theorem 7 then the quaternion matrix equations 1198603119883119860lowast
3=
1198623and 119860
4119883119860lowast
4= 1198624have common Hermitian solution if and
only if (30) hold and the following equalities are satisfied
119903 [1198603 1198623] = 119903 (1198603)
119903 [
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
= 2119903 [1198603
1198604
]
(38)
Corollary 9 Suppose that11986011198621isin H119898times119899119861
11198622isin H119899times119904 and
119860 119861 isin H119899times119899 are Hermitian Then 119860 and 119861 have a common
Hermitian g-inverse which is a solution to the system (2) if andonly if (11) holds and the following equalities are all satisfied
119903[[
[
11986011198621119860
119861lowast
1119862lowast
2119860
119860 119860
]]
]
= 119903[
[
1198601
119861lowast
1
119860
]
]
119903[[
[
11986011198621119861
119861lowast
1119862lowast
2119861
119861 119861
]]
]
= 119903[
[
1198601
119861lowast
1
119861
]
]
(39)
119903
[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861 119861 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]
]
= 2119903
[[[[
[
119861
119860
119861lowast
1
1198601
]]]]
]
(40)
4 Extremal Ranks of Schur ComplementSubject to (2)
As is well known for a given block matrix
119872 = [119860 119861
119861lowast119863] (41)
where 119860 and 119863 are Hermitian quaternion matrices withappropriate sizes then the Hermitian Schur complement of119860 in119872 is defined as
119878119860= 119863 minus 119861
lowast119860sim119861 (42)
where 119860sim is a Hermitian g-inverse of 119860 that is 119860sim isin 119883 |
119860119883119860 = 119860119883 = 119883lowast
Now we use Theorem 4 to establish the extremal ranksof 119878119860given by (42) with respect to 119860sim which is a solution to
system (2)
Theorem 10 Suppose 1198601 1198621isin H119898times119899 119861
1 1198622isin H119899times119904 119863 isin
H119905times119905 119861 isin H119899times119905 and 119860 isin H119899times119899 are given and system (2)is consistent then the extreme ranks of 119878
119860given by (42) with
respect to 119860sim which is a solution of (2) are the following
max1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = min 119886 119887
(43)
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
from
[[[[[[
[
119868 0 0 0 0
0 119868 0 0 0
119860311988300 119868 0 0
119861lowast
111988300 0 119868 0
119860111988300 0 0 119868
]]]]]]
]
times
[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
times
[[[[[
[
119868 minus1198830119860lowast
40 0 0
0 119868 0 0 0
0 0 119868 0 0
0 0 0 119868 0
0 0 0 0 119868
]]]]]
]
=
[[[[[
[
0 119860lowast
4119860lowast
31198611119860lowast
1
1198604
0 0 0 0
1198603
0 0 0 0
119861lowast
10 0 0 0
1198601
0 0 0 0
]]]]]
]
(34)
that (32) holds Similarly we can obtain (31)Conversely assume that 119862
3= 119862lowast
3 (11) (13) hold then by
Lemma 1 system (10) have Hermitian solution By (20) (31)-(32) and
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
ge 119903[[[
[
1198604
1198603
119861lowast
1
1198601
]]]
]
+ 119903[
[
1198604
119861lowast
1
1198601
]
]
(35)
we can get
min 119903 [119891 (119883)] le 0 (36)
However
min 119903 [119891 (119883)] ge 0 (37)
Hence (33) holds implying that the system (5) have Hermi-tian solution
ByTheorem 7 we can also get the following
Corollary 8 Suppose that 1198603 1198623 1198604 and 119862
4are those in
Theorem 7 then the quaternion matrix equations 1198603119883119860lowast
3=
1198623and 119860
4119883119860lowast
4= 1198624have common Hermitian solution if and
only if (30) hold and the following equalities are satisfied
119903 [1198603 1198623] = 119903 (1198603)
119903 [
[
0 119860lowast
4119860lowast
3
11986041198624
0
1198603
0 minus1198623
]
]
= 2119903 [1198603
1198604
]
(38)
Corollary 9 Suppose that11986011198621isin H119898times119899119861
11198622isin H119899times119904 and
119860 119861 isin H119899times119899 are Hermitian Then 119860 and 119861 have a common
Hermitian g-inverse which is a solution to the system (2) if andonly if (11) holds and the following equalities are all satisfied
119903[[
[
11986011198621119860
119861lowast
1119862lowast
2119860
119860 119860
]]
]
= 119903[
[
1198601
119861lowast
1
119860
]
]
119903[[
[
11986011198621119861
119861lowast
1119862lowast
2119861
119861 119861
]]
]
= 119903[
[
1198601
119861lowast
1
119861
]
]
(39)
119903
[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861 119861 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]
]
= 2119903
[[[[
[
119861
119860
119861lowast
1
1198601
]]]]
]
(40)
4 Extremal Ranks of Schur ComplementSubject to (2)
As is well known for a given block matrix
119872 = [119860 119861
119861lowast119863] (41)
where 119860 and 119863 are Hermitian quaternion matrices withappropriate sizes then the Hermitian Schur complement of119860 in119872 is defined as
119878119860= 119863 minus 119861
lowast119860sim119861 (42)
where 119860sim is a Hermitian g-inverse of 119860 that is 119860sim isin 119883 |
119860119883119860 = 119860119883 = 119883lowast
Now we use Theorem 4 to establish the extremal ranksof 119878119860given by (42) with respect to 119860sim which is a solution to
system (2)
Theorem 10 Suppose 1198601 1198621isin H119898times119899 119861
1 1198622isin H119899times119904 119863 isin
H119905times119905 119861 isin H119899times119905 and 119860 isin H119899times119899 are given and system (2)is consistent then the extreme ranks of 119878
119860given by (42) with
respect to 119860sim which is a solution of (2) are the following
max1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = min 119886 119887
(43)
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
where
119886 = 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
minus 119903 [119861lowast
1
1198601
]
119887 = 119903
[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
minus 2119903[
[
119860
119861lowast
1
1198601
]
]
min1198601119860sim
=1198621
119860sim
1198611=1198622
119903 (119878119860) = 2119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
minus 2119903
[[[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
(44)
Proof It is obvious that
max1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= max1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
min1198601119860sim=1198621119860sim1198611=1198622
119903 (119863 minus 119861lowast119860sim119861)
= min1198601119883=11986211198831198611=1198622119860119883119860=119860
119903 (119863 minus 119861lowast119883119861)
(45)
Thus in Theorem 4 and its proof letting 1198603= 119860lowast
3= 1198623= 119860
1198604= 119861lowast and 119862
4= 119863 we can easily get the proof
In Theorem 10 let 1198601 1198621 1198611 and 119862
2vanish Then we
can easily get the following
Corollary 11 The extreme ranks of 119878119860given by (42) with
respect to 119860sim are the following
max119860sim
119903 (119878119860) = min
119903 [119863 119861lowast] 119903 [
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903 (119860)
min119860sim
119903 (119878119860) = 2119903 [119863 119861
lowast] + 119903[
[
0 119861 119860
119861lowast119863 0
119860 0 minus119860
]
]
minus 2119903[
[
0 119861
119861lowast119863
119860 0
]
]
(46)
5 The Rank Invariance of (9)As another application of Theorem 4 we in this sectionconsider the rank invariance of thematrix expression (9) withrespect to the Hermitian solution of system (10)
Theorem 12 Suppose that (10) have Hermitian solution thenthe rank of 119891(119883) defined by (9) with respect to the Hermitiansolution of (10) is invariant if and only if
119903
[[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119860lowast
4119860lowast
31198611
119860lowast
1
11986041198624
0 0 0
1198603
0 minus1198623
minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
2119860lowast
3minus119862lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus1198621119860lowast
3minus11986211198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
1198603
119861lowast
1
1198601
]
]
(47)
or
119903
[[[[[[[
[
0 119860lowast
41198611
119860lowast
1
11986041198624
0 0
1198603
0 minus11986031198622minus1198603119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
1198601
0 minus11986211198611minus1198621119860lowast
1
]]]]]]]
]
= 119903[[
[
1198624
1198604
119862lowast
2119860lowast
4119861lowast
1
1198621119860lowast
41198601
]]
]
+ 119903[[
[
1198603
119861lowast
1
1198601
]]
]
(48)
Proof It is obvious that the rank of 119891(119883) with respect toHermitian solution of system (10) is invariant if and only if
max 119903 [119891 (119883)] minusmin 119903 [119891 (119883)] = 0 (49)
By (49) Theorem 4 and simplifications we can get (47)and (48)
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
Corollary 13 The rank of 119878119860defined by (42) with respect to
119860sim which is a solution to system (2) is invariant if and only if
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
119903
[[[[[[[[
[
0 119861 119860 1198611
119860lowast
1
119861lowast119863 0 0 0
119860 0 minus119860 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
2119860 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
1119860 minus119862
11198611minus1198621119860lowast
1
]]]]]]]]
]
+ 119903 [119861lowast
1
1198601
]
= 119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(50)
or
119903
[[[[[[
[
0 119861 1198611
119860lowast
1
119861lowast119863 0 0
119860 0 minus1198601198622
minus119860119862lowast
1
119861lowast
10 minus119862
lowast
21198611minus119862lowast
2119860lowast
1
11986010 minus119862
11198611minus1198621119860lowast
1
]]]]]]
]
= 119903[
[
119863 119861lowast
119862lowast
2119861 119861lowast
1
1198621119861 1198601
]
]
+ 119903[
[
119860
119861lowast
1
1198601
]
]
(51)
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China Tian Yuan Foundation (11226067) theFundamental Research Funds for the Central Universities(WM1214063) and China Postdoctoral Science Foundation(2012M511014)
References
[1] T W Hungerford Algebra Springer New York NY USA 1980[2] S K Mitra ldquoThe matrix equations 119860119883 = 119862 119883119861 = 119863rdquo Linear
Algebra and its Applications vol 59 pp 171ndash181 1984[3] C G Khatri and S K Mitra ldquoHermitian and nonnegative
definite solutions of linear matrix equationsrdquo SIAM Journal onApplied Mathematics vol 31 no 4 pp 579ndash585 1976
[4] Y X Yuan ldquoOn the symmetric solutions of matrix equation(119860119883119883119862) = (119861119863)rdquo Journal of East China Shipbuilding Institutevol 15 no 4 pp 82ndash85 2001 (Chinese)
[5] H Dai and P Lancaster ldquoLinear matrix equations from aninverse problem of vibration theoryrdquo Linear Algebra and itsApplications vol 246 pp 31ndash47 1996
[6] J Groszlig ldquoA note on the general Hermitian solution to 119860119883119860lowast =119861rdquo Malaysian Mathematical Society vol 21 no 2 pp 57ndash621998
[7] Y G Tian and Y H Liu ldquoExtremal ranks of some symmetricmatrix expressions with applicationsrdquo SIAM Journal on MatrixAnalysis and Applications vol 28 no 3 pp 890ndash905 2006
[8] Y H Liu Y G Tian and Y Takane ldquoRanks of Hermitian andskew-Hermitian solutions to the matrix equation 119860119883119860lowast = 119861rdquoLinear Algebra and its Applications vol 431 no 12 pp 2359ndash2372 2009
[9] X W Chang and J S Wang ldquoThe symmetric solution of thematrix equations 119860119883 + 119884119860 = 119862 119860119883119860119879 + 119861119884119861
119879= 119862 and
(119860119879119883119860 119861
119879119883119861) = (119862119863)rdquo Linear Algebra and its Applications
vol 179 pp 171ndash189 1993[10] Q-W Wang and Z-C Wu ldquoCommon Hermitian solutions
to some operator equations on Hilbert 119862lowast-modulesrdquo LinearAlgebra and its Applications vol 432 no 12 pp 3159ndash3171 2010
[11] F O Farid M S Moslehian Q-W Wang and Z-C Wu ldquoOnthe Hermitian solutions to a system of adjointable operatorequationsrdquo Linear Algebra and its Applications vol 437 no 7pp 1854ndash1891 2012
[12] Z-H He and Q-WWang ldquoSolutions to optimization problemson ranks and inertias of a matrix function with applicationsrdquoAppliedMathematics andComputation vol 219 no 6 pp 2989ndash3001 2012
[13] Q-W Wang ldquoThe general solution to a system of real quater-nion matrix equationsrdquo Computers amp Mathematics with Appli-cations vol 49 no 5-6 pp 665ndash675 2005
[14] Q-W Wang ldquoBisymmetric and centrosymmetric solutions tosystems of real quaternion matrix equationsrdquo Computers ampMathematics with Applications vol 49 no 5-6 pp 641ndash6502005
[15] Q W Wang ldquoA system of four matrix equations over vonNeumann regular rings and its applicationsrdquo Acta MathematicaSinica vol 21 no 2 pp 323ndash334 2005
[16] Q-W Wang ldquoA system of matrix equations and a linear matrixequation over arbitrary regular rings with identityrdquo LinearAlgebra and its Applications vol 384 pp 43ndash54 2004
[17] Q-WWang H-X Chang andQNing ldquoThe common solutionto six quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 198 no 1 pp 209ndash2262008
[18] Q-W Wang H-X Chang and C-Y Lin ldquoP-(skew)symmetriccommon solutions to a pair of quaternion matrix equationsrdquoApplied Mathematics and Computation vol 195 no 2 pp 721ndash732 2008
[19] Q W Wang and Z H He ldquoSome matrix equations withapplicationsrdquo Linear and Multilinear Algebra vol 60 no 11-12pp 1327ndash1353 2012
[20] Q W Wang and J Jiang ldquoExtreme ranks of (skew-)Hermitiansolutions to a quaternion matrix equationrdquo Electronic Journal ofLinear Algebra vol 20 pp 552ndash573 2010
[21] Q-W Wang and C-K Li ldquoRanks and the least-norm of thegeneral solution to a system of quaternion matrix equationsrdquoLinear Algebra and its Applications vol 430 no 5-6 pp 1626ndash1640 2009
[22] Q-W Wang X Liu and S-W Yu ldquoThe common bisymmetricnonnegative definite solutions with extreme ranks and inertiasto a pair of matrix equationsrdquo Applied Mathematics and Com-putation vol 218 no 6 pp 2761ndash2771 2011
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
[23] Q-W Wang F Qin and C-Y Lin ldquoThe common solution tomatrix equations over a regular ring with applicationsrdquo IndianJournal of Pure andAppliedMathematics vol 36 no 12 pp 655ndash672 2005
[24] Q-W Wang G-J Song and C-Y Lin ldquoExtreme ranks of thesolution to a consistent system of linear quaternion matrixequations with an applicationrdquo Applied Mathematics and Com-putation vol 189 no 2 pp 1517ndash1532 2007
[25] Q-W Wang G-J Song and C-Y Lin ldquoRank equalities relatedto the generalized inverse 119860
(2)
119879119878with applicationsrdquo Applied
Mathematics and Computation vol 205 no 1 pp 370ndash3822008
[26] Q W Wang G J Song and X Liu ldquoMaximal and minimalranks of the common solution of some linear matrix equationsover an arbitrary division ring with applicationsrdquo AlgebraColloquium vol 16 no 2 pp 293ndash308 2009
[27] Q-W Wang Z-C Wu and C-Y Lin ldquoExtremal ranks ofa quaternion matrix expression subject to consistent systemsof quaternion matrix equations with applicationsrdquo AppliedMathematics and Computation vol 182 no 2 pp 1755ndash17642006
[28] Q W Wang and S W Yu ldquoRanks of the common solution tosome quaternion matrix equations with applicationsrdquo Bulletinof Iranian Mathematical Society vol 38 no 1 pp 131ndash157 2012
[29] Q W Wang S W Yu and W Xie ldquoExtreme ranks of realmatrices in solution of the quaternion matrix equation 119860119883119861 =
119862with applicationsrdquoAlgebra Colloquium vol 17 no 2 pp 345ndash360 2010
[30] Q-W Wang S-W Yu and Q Zhang ldquoThe real solutions toa system of quaternion matrix equations with applicationsrdquoCommunications in Algebra vol 37 no 6 pp 2060ndash2079 2009
[31] Q-W Wang S-W Yu and C-Y Lin ldquoExtreme ranks of alinear quaternionmatrix expression subject to triple quaternionmatrix equations with applicationsrdquo Applied Mathematics andComputation vol 195 no 2 pp 733ndash744 2008
[32] Q-W Wang and F Zhang ldquoThe reflexive re-nonnegativedefinite solution to a quaternion matrix equationrdquo ElectronicJournal of Linear Algebra vol 17 pp 88ndash101 2008
[33] Q W Wang X Zhang and Z H He ldquoOn the Hermitianstructures of the solution to a pair of matrix equationsrdquo Linearand Multilinear Algebra vol 61 no 1 pp 73ndash90 2013
[34] X Zhang Q-W Wang and X Liu ldquoInertias and ranks ofsome Hermitian matrix functions with applicationsrdquo CentralEuropean Journal of Mathematics vol 10 no 1 pp 329ndash3512012
[35] Y G Tian and S Z Cheng ldquoThemaximal andminimal ranks of119860 minus 119861119883119862 with applicationsrdquo New York Journal of Mathematicsvol 9 pp 345ndash362 2003
[36] Y G Tian ldquoUpper and lower bounds for ranks of matrixexpressions using generalized inversesrdquo Linear Algebra and itsApplications vol 355 pp 187ndash214 2002
[37] Y H Liu and Y G Tian ldquoMore on extremal ranks of thematrix expressions119860minus119861119883plusmn119883
lowast119861lowast with statistical applicationsrdquo
Numerical Linear Algebra with Applications vol 15 no 4 pp307ndash325 2008
[38] Y H Liu and Y G Tian ldquoMax-min problems on the ranksand inertias of the matrix expressions 119860 minus 119861119883119862 plusmn (119861119883119862)
lowast withapplicationsrdquo Journal of Optimization Theory and Applicationsvol 148 no 3 pp 593ndash622 2011
[39] T Ando ldquoGeneralized Schur complementsrdquo Linear Algebra andits Applications vol 27 pp 173ndash186 1979
[40] D Carlson E Haynsworth and T Markham ldquoA generalizationof the Schur complement by means of the Moore-Penroseinverserdquo SIAM Journal onAppliedMathematics vol 26 pp 169ndash175 1974
[41] M Fiedler ldquoRemarks on the Schur complementrdquo Linear Algebraand its Applications vol 39 pp 189ndash195 1981
[42] Y G Tian ldquoMore on maximal and minimal ranks of Schurcomplements with applicationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 675ndash692 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![arXiv:math/0003224v1 [math.RA] 30 Mar 2000 · rank equalities for matrix expressions involving Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose](https://static.fdocuments.us/doc/165x107/5f961236b05e766be7258ab6/arxivmath0003224v1-mathra-30-mar-2000-rank-equalities-for-matrix-expressions.jpg){kind=icon}
