Research Article Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent...
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Research Article Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products Li Yang, 1 Wei Zhang, 2 and Jinli Xu 3 1 Department of Foundation, Harbin Finance University, Harbin 150030, China 2 School of Mathematical Science, Heilongjiang University, Harbin 150080, China 3 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Correspondence should be addressed to Wei Zhang; [email protected] Received 26 October 2013; Accepted 10 December 2013; Published 28 January 2014 Academic Editor: Antonio M. Peralta Copyright © 2014 Li Yang et al. 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. Suppose , ≥ 2 are positive integers. Let T be the space of all × complex upper triangular matrices, and let be an injective linear map on T ⊗ T . en ( ⊗ ) is an idempotent matrix in T ⊗ T whenever ⊗ is an idempotent matrix in T ⊗ T if and only if there exists an invertible matrix ∈ T ⊗ T such that ( ⊗ ) = ( 1 () ⊗ 2 ()) −1 , ∀ ∈ T ,∈ T , or when =, ( ⊗ ) = ( 1 () ⊗ 2 ()) −1 , ∀ ∈ T ,∈ T , where 1 ([ ]) = [ ] or 1 ([ ]) = [ −+1,−+1 ] and 2 ([ ]) = [ ] or 2 ([ ]) = [ −+1,−+1 ]. 1. Introduction Suppose , ≥ 2 are positive integers. Let M be the space of all × complex matrices, and let T be all upper triangular in M . For ∈ M , ∈ M , we denote by ⊗ their tensor product (a.k.a. Kronecker product). Linear preserver problem is a hot area in matrix and operator theory; there are many results about this area (see [1– 14]). Specially, the idempotence preservers and the rank one preservers play an important role (see [1, 2]); therefore, it is meaningful to study the idempotence preservers. Chan et al. [3] first characterize linear transformations on M preserving idempotent matrices. ˇ Semrl [4] applying projective geometry gives the form of transformations on rank-1 idempotents. Tang et al. [5] investigate injective linear idempotence pre- servers on T . In quantum information science, quantum states of a system with physical states are represented as density matrices, that is, positive semidefinite matrices with trace one. If ∈ M and ∈ M are two quantum states in two quantum systems, then ⊗ describes a joint state in bipartite system M ⊗ M . Recently, many researchers con- sider the problem combining linear preserver problem with quantum information science. ey determine the structure of linear maps on M ⊗ M by using information only about the images of matrices possessing tensor product form. One can see [15–18] and their references for some background on linear preserver problems on tensor spaces arising in quantum information science. Inspired by the above, the purpose of this paper is to study injective linear maps on T ⊗ T satisfying ( ⊗ ) is an idempotent matrix whenever ⊗ is an idempotent matrix in T ⊗ T . If we remove the assumption that map is injective, then may have various forms as follows. Example 1. ⊗ → 11 ⊗ is a linear idempotent preserver on T ⊗ T . Example 2. ⊗ → 11 ( () 11 + () 22 ) ⊗ ( 11 () 11 + 12 () 12 + 22 () 22 ) is a linear idempotent preserver on T ⊗ T . We end this section by introducing some notations which will be used in the following sections. Let C be the complex field, the × identity matrix, 0 the zero matrix whose order is omitted in different matrices just for simplicity, and (resp., rank ) the transpose (resp., rank) of . () , Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 148321, 8 pages http://dx.doi.org/10.1155/2014/148321
Transcript of Research Article Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent...
Research ArticleLinear Maps on Upper TriangularMatrices Spaces Preserving Idempotent Tensor Products
Li Yang1 Wei Zhang2 and Jinli Xu3
1 Department of Foundation Harbin Finance University Harbin 150030 China2 School of Mathematical Science Heilongjiang University Harbin 150080 China3Department of Mathematics Harbin Institute of Technology Harbin 150001 China
Correspondence should be addressed to Wei Zhang weizhanghlj163com
Received 26 October 2013 Accepted 10 December 2013 Published 28 January 2014
Academic Editor Antonio M Peralta
Copyright copy 2014 Li Yang et al This 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
Suppose119898 119899 ge 2 are positive integers LetT119899be the space of all 119899 times 119899 complex upper triangular matrices and let 120601 be an injective
linear map onT119898otimesT119899 Then 120601(119860otimes119861) is an idempotent matrix inT
119898otimesT119899whenever119860otimes119861 is an idempotent matrix inT
119898otimesT119899
if and only if there exists an invertible matrix 119875 isin T119898otimesT119899such that 120601(119860otimes119861) = 119875(120585
1(119860)otimes120585
2(119861))119875
minus1
forall119860 isin T119898 119861 isin T
119899 or when
119898 = 119899 120601(119860 otimes 119861) = 119875(1205851(119861) otimes 120585
2(119860))119875
minus1
forall119860 isin T119898 119861 isin T
119898 where 120585
1([119886119894119895]) = [119886
119894119895] or 1205851([119886119894119895]) = [119886
119898minus119894+1119898minus119895+1] and 120585
2([119887119894119895]) = [119887
119894119895]
or 1205852([119887119894119895]) = [119887
119899minus119894+1119899minus119895+1]
1 Introduction
Suppose119898 119899 ge 2 are positive integers LetM119899be the space of
all 119899 times 119899 complex matrices and letT119899be all upper triangular
inM119899 For119860 isin M
119898 119861 isin M
119899 we denote by119860otimes119861 their tensor
product (aka Kronecker product)Linear preserver problem is a hot area in matrix and
operator theory there aremany results about this area (see [1ndash14]) Specially the idempotence preservers and the rank onepreservers play an important role (see [1 2]) therefore it ismeaningful to study the idempotence preservers Chan et al[3] first characterize linear transformations onM
119899preserving
idempotent matrices Semrl [4] applying projective geometrygives the form of transformations on rank-1 idempotentsTang et al [5] investigate injective linear idempotence pre-servers onT
119899
In quantum information science quantum states of asystem with 119899 physical states are represented as densitymatrices that is positive semidefinite matrices with traceone If 119860 isin M
119898and 119861 isin M
119899are two quantum states in
two quantum systems then 119860 otimes 119861 describes a joint state inbipartite systemM
119898otimesM119899 Recently many researchers con-
sider the problem combining linear preserver problem with
quantum information science They determine the structureof linear maps on M
119898otimes M119899by using information only
about the images of matrices possessing tensor product formOne can see [15ndash18] and their references for some backgroundon linear preserver problems on tensor spaces arising inquantum information science
Inspired by the above the purpose of this paper is to studyinjective linear maps 120601 on T
119898otimes T119899satisfying 120601(119860 otimes 119861)
is an idempotent matrix whenever 119860 otimes 119861 is an idempotentmatrix inT
119898otimesT119899 If we remove the assumption that map is
injective then 120601may have various forms as follows
Example 1 119860 otimes 119861 997891rarr 11988611119868119898
otimes 119861 is a linear idempotentpreserver onT
119898otimesT119899
Example 2 119860 otimes 119861 997891rarr 11988611(119864(119898)
11+ 119864(119898)
22) otimes (11988711119864(119899)
11+ 11988712119864(119899)
12+
11988722119864(119899)
22) is a linear idempotent preserver onT
119898otimesT119899
We end this section by introducing some notations whichwill be used in the following sections Let C be the complexfield 119868
119896the 119896 times 119896 identity matrix 0 the zero matrix whose
order is omitted in different matrices just for simplicity and119883119905 (resp rank119883) the transpose (resp rank) of 119883 119864(119899)
119894119895
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 148321 8 pageshttpdxdoiorg1011552014148321
2 Abstract and Applied Analysis
forall119894 119895 isin [1 119899] stands for the 119899 times 119899 matrix with 1 at the(119894 119895)th entry and 0 otherwise Denote by 119869
119896(also 119869) thematrix
119864(119896)
1119896+ 119864(119896)
2119896minus1+ sdot sdot sdot + 119864
(119896)
1198961 Clearly if 119860 = [119886
119894119895] isin T
119899 then
119869119860119905
119869 = [119886119899minus119895+1119899minus119894+1
] isin T119899 For positive integers 119899
1and 119899
2
with 1198991lt 1198992 let [119899
1 1198992] be the set of all integers between 119899
1
and 1198992 For any (119894 119895) isin [1119898] times [1 119899] we define 120588 by
120588 (119894 119895) = (119894 minus 1) 119899 + 119895 (1)
For any 119896 isin [1119898119899] we define 120590(119896) isin [1119898] and 120591 (119896) isin [1 119899]such that 119896 = (120590(119896) minus 1)119899 + 120591(119896) (it is easy to see that 120590 and 120591are well defined) It is easy to see that
119864(119898)
119903119904otimes 119864(119899)
119906V = 119864(119898119899)
(119903minus1)119899+119906(119904minus1)119899+V = 119864(119898119899)
120588(119903119906)120588(119904V) (2)
119864(119898119899)
119894119895= 119864(119898)
120590(119894)120590(119895)otimes 119864(119899)
120591(119894)120591(119895) (3)
We define a partial ordering of [1 119898]times[1 119899] by (119886 119887) le (119888 119889)
if and only if 119886 le 119888 and 119887 le 119889 We say that (119886 119887) and (119888 119889) arecomparable if (119886 119887) le (119888 119889) or (119888 119889) le (119886 119887)
2 Preliminary Results
We need the form of injective linear idempotence preserveronT119899 which was obtained in [5]
Lemma 3 (see [5 Theorem 1]) Let 120595 be an injective linearmap on T
119899 Then 120595 (119883) is an idempotent matrix in T
119899
whenever119883 is an idempotent matrix inT119899if and only if there
exists an invertible matrix 119875 isin T119899such that
120595 (119883) = 119875120585 (119883) 119875minus1
forall119883 isin T119899 (4)
where 120585(119883) = 119883 or 120585(119883) = 119869119883119905
119869
It is clear thatT119898otimesT119899⊊ T119898119899 For example 119864(4)
23isin T4
and
119864(4)
23notin T2otimesT2=
[
[
[
[
[
[
[
[
[
11988611
11988612
11988613
11988614
0 11988622
0 11988624
0 0 11988633
11988634
0 0 0 11988644
]
]
]
]
]
]
]
]
]
119886119894119895isin C
(5)
It is easy to see that (120590(2) 120591(2)) = (1 2) and (120590(3) 120591(3)) =
(2 1) In fact we can point out the positions of elementswhich are inT
119898119899T119898otimesT119899
Lemma 4 120582119864(119898119899)119894119895
isin T119898otimes T119899if and only if (120590(119894) 120591(119894)) le
(120590(119895) 120591(119895)) or 120582 = 0
Proof It is a direct corollary of (3)
The next Lemma describes the partial ordering wedefined in Section 1 which is useful to prove our mainTheorem
Lemma 5 (see [19 Theorem 1]) Let 119898 119899 ge 2 and let 119860 bea matrix with 119898 rows and 119899 columns containing all elementsof [1 119898] times [1 119899] If every two elements of 119860 in the samerow and column are comparable respectively then there existpermutation matrices 119880 isin M
119898and 119881 isin M
119899such that
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
[
[
(1 1) (1 2) sdot sdot sdot (1 119899 minus 1) (1 119899)
(2 1) (2 2) sdot sdot sdot (2 119899 minus 1) (2 119899)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(119898 minus 1 1) (119898 minus 1 2) sdot sdot sdot (119898 minus 1 119899 minus 1) (119898 minus 1 119899)
(119898 1) (119898 2) sdot sdot sdot (119898 119899 minus 1) (119898 119899)
]
]
]
]
]
]
]
]
]
]
]
]
]
(I)
or when119898 ge 3 or 119899 ge 3
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
(119898 119899) (1 2) sdot sdot sdot (1 119899 minus 1) (1 119899)
(2 1) (2 2) sdot sdot sdot (2 119899 minus 1) (2 119899)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(119898 minus 1 1) (119898 minus 1 2) sdot sdot sdot (119898 minus 1 119899 minus 1) (119898 minus 1 119899)
(119898 1) (119898 2) sdot sdot sdot (119898 119899 minus 1) (1 1)
]
]
]
]
]
]
]
]
]
]
]
(II)
or when119898 = 119899
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
(1 1) (2 1) sdot sdot sdot (119898 minus 1 1) (119898 1)
(1 2) (2 2) sdot sdot sdot (119898 minus 1 2) (119898 2)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(1 119898 minus 1) (2119898 minus 1) sdot sdot sdot (119898 minus 1119898 minus 1) (119898119898 minus 1)
(1119898) (2119898) sdot sdot sdot (119898 minus 1119898) (119898119898)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(III)
or when119898 = 119899 ge 3
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
[
(119898119898) (2 1) sdot sdot sdot (119898 minus 1 1) (119898 1)
(1 2) (2 2) sdot sdot sdot (119898 minus 1 2) (119898 2)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(1 119898 minus 1) (2119898 minus 1) sdot sdot sdot (119898 minus 1119898 minus 1) (119898119898 minus 1)
(1119898) (2119898) sdot sdot sdot (119898 minus 1119898) (1 1)
]
]
]
]
]
]
]
]
]
]
]
]
(IV)
Abstract and Applied Analysis 3
The following lemmas would make the proof of the maintheorem more concise
Lemma6 Suppose119883 isin M119899is an idempotentmatrix such that
0119904oplus119868119903oplus0119899minus119903minus119904
minus119883 also is an idempotentmatrixThen there existsan idempotent matrix119883
119903isin M119903such that119883 = 0
119904oplus119883119903oplus0119899minus119903minus119904
Proof By 1198832 = 119883 and (0119904oplus 119868119903oplus 0119899minus119903minus119904
minus 119883)2
= 0119904oplus 119868119903oplus
0119899minus119903minus119904
minus 119883 we have
2119883 = (0119904oplus 119868119903oplus 0119899minus119903minus119904
)119883 + 119883 (0119904oplus 119868119903oplus 0119899minus119903minus119904
) (6)
Set
119883 =
[
[
[
[
[
11988311
11988312
11988313
11988321
11988322
11988323
11988331
11988332
11988333
]
]
]
]
]
(7)
where11988311isin M119904 11988322isin M119903 Then (6) implies
2
[
[
[
[
[
11988311
11988312
11988313
11988321
11988322
11988323
11988331
11988332
11988333
]
]
]
]
]
=
[
[
[
[
[
0 11988312
0
11988321
211988322
11988323
0 11988332
0
]
]
]
]
]
(8)
Hence 119883 = 0119904oplus 11988322
oplus 0119899minus119903minus119904
therefore the lemma holds
Lemma 7 Let 119883 isin T119898otimesT119899be an idempotent matrix such
that 119864(119898)119894119894
otimes 119868119899minus 119883 also is an idempotent matrix Then there
exists an idempotent 119884 isin T119899such that 119883 = 119864
(119898)
119894119894otimes 119884
Proof The proof is similar to that of Lemma 6
Lemma 8 Suppose 119903 119904 isin [1 119899 minus 1] If for any 120582 isin C 119868119903oplus
0119899minus119903
+120582119883 and 0119903oplus119868119904oplus0119899minus119903minus119904
+120582119883 are idempotent inT119899 then
X = [
01199031198831
0 0119904
] oplus 0119899minus119903minus119904
(9)
Proof It follows from 119868119903oplus 0119899minus119903
+ 120582119883 forall120582 isin C is idempotentthat
(119868119903oplus 0119899minus119903
)119883 + 119883 (119868119903oplus 0119899minus119903
) = 119883 (10)
Let
119883 =[
[
[
1198831199031198831
1198832
0 119883119904
1198833
0 0 119883119899minus119903minus119904
]
]
]
(11)
where119883119903isin T119903119883119904isin T119904 then (10) implies
[
[
[
[
[
211988311990311988311198832
0 0 0
0 0 0
]
]
]
]
]
=
[
[
[
[
[
1198831199031198831
1198832
0 119883119904
1198833
0 0 119883119899minus119903minus119904
]
]
]
]
]
(12)
Hence 119883119903= 0 119883
119904= 0 119883
119899minus119903minus119904= 0 119883
3= 0 Similarly from
0119903oplus 119868119904oplus 0119899minus119903minus119904
+ 120582119883 being idempotent we have1198832= 0
Lemma 9 (see [6 Page 62 Exercise 1]) Suppose1198601 119860
119896isin
M119899are idempotent matrices such that for any 119894 = 119895 isin [1 119896]
119860119894+ 119860119895is idempotent Let 119903
119894= rank119860
119894 Then there exists an
invertible matrix 119875 isin M119899such that
119860119894= 119875 diag (0 0 1 1 0 0) 119875minus1 (13)
where diag (0 0 1 1 0 0) is the diagonalmatrix inwhich all diagonal entries are zero except those in the (119903
1+sdot sdot sdot+
119903119894minus1
+ 1)st to the (1199031+ sdot sdot sdot + 119903
119894)th rows
Similar to Lemma 9 we have the following
Lemma 10 Let 1198601 119860
119898119899isin T119898otimes T119899be idempotent
matrices of rank-1 such that for any 119894 = 119895 isin [1119898119899] 119860119894+ 119860119895is
idempotent Then there exist a permutation 120587 on [1 119898119899] andan invertible matrix 119875 isin T
119898otimesT119899such that
119860119894= 119875119864(119898119899)
120587(119894)120587(119894)119875minus1
119894 = 1 119898119899 (14)
Proof By119860119894isin T119898otimesT119899being an idempotentmatrix of rank-
1 we can assume
119860119894= 119864(119898119899)
120587(119894)120587(119894)+ 119861119894 119894 = 1 119898119899 (15)
where 119861119894isin T119898otimes T119899with zero diagonal entries It follows
from 119860119894+ 119860119895 forall119894 = 119895 being is idempotent that 120587(119894) = 120587(119895)
forall119894 = 119895 Hence 120587 is a permutation on [1 119898119899] By 1198602120587minus1
(1)=
119860120587minus1
(1) we can see 119860
120587minus1
(1)= 119864(119898119899)
11+ Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896 Let
1198751= 119868119898119899
minus Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896isin T119898otimes T119899 then 119875
minus1
1= 119868119898119899
+
Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896and
119860120587minus1
(1)= 1198751119864(119898119899)
11119875minus1
1 (16)
By 119860120587minus1
(1)+ 119860
120587minus1
(2)being idempotent we obtain
119875minus1
1119860120587minus1
(2)1198751
= 119864(119898119899)
22+ Σ(119898119899)
119896=31205822119896119864(119898119899)
2119896 Let 119875
2=
119868119898119899
minus Σ(119898119899)
119896=31205822119896119864(119898119899)
2119896 then
119860120587minus1
(1)= 11987521198751119864(119898119899)
11119875minus1
1119875minus1
2
119860120587minus1
(2)= 11987521198751119864(119898119899)
22119875minus1
1119875minus1
2
(17)
Continuing to do this we can find 1198753 119875
119898119899 Let 119875 =
119875119898119899119875119898119899minus1
sdot sdot sdot 11987521198751isin T119898otimesT119899 then
119860120587minus1
(119894)= 119875119864(119898119899)
119894119894119875minus1
119894 = 1 119898119899 (18)
This completes the proof
3 The Main Result
Themain result of this paper is as follows
Theorem 11 Suppose 119898 119899 ge 2 are positive integers and 120601
is an injective linear map on T119898otimes T119899 Then 120601 (119860 otimes 119861)
4 Abstract and Applied Analysis
is an idempotent matrix in T119898otimes T119899whenever 119860 otimes 119861 is an
idempotent matrix in T119898otimes T119899if and only if there exists an
invertible matrix 119875 isin T119898otimesT119899such that
120601 (119860 otimes 119861) = 119875 (1205851(119860) otimes 120585
2(119861)) 119875
minus1
forall119860 isin T119898 119861 isin T
119899
(i)
or when119898 = 119899
120601 (119860 otimes 119861) = 119875 (1205851(119861) otimes 120585
2(119860)) 119875
minus1
forall119860 isin T119898 119861 isin T
119898
(ii)
where for 119894 = 1 2 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869
Proof The sufficiency is obvious We will prove the necessityby the following six steps
Step 1 There exist a permutation 120587 on [1 119898119899] and aninvertible matrix 119875 isin T
119898otimesT119899such that
120601 (119864(119898119899)
119896119896) = 119875119864
(119898119899)
120587(119896)120587(119896)119875minus1
forall119896 isin [1119898119899] (19)
Proof of Step 1 By Lemma 10 we only need to prove that
rank 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) = 1 forall119894 isin [1119898] 119895 isin [1 119899] (20)
And for any (119894 119895) = (119906 V) isin [1119898] times [1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119906119906otimes 119864(119899)
VV ) is an idempotentmatrix(21)
It follows from 119864(119898)
11otimes 119868119899 119864
(119898)
119898119898otimes 119868119899and (119864(119898)
119894119894+119864(119898)
119895119895) otimes
119868119899forall119894 = 119895 isin [1119898] are idempotent matrices in T
119898otimes T119899that
120601 (119864(119898)
11otimes119868119899) 120601 (119864
(119898)
119898119898otimes119868119899) and 120601 (119864(119898)
119894119894otimes119868119899)+120601 (119864
(119898)
119895119895otimes119868119899)
forall119894 = 119895 isin [1119898] are idempotent matrices We obtain by usingLemma 9 that there exists an invertible matrix 119875
1isin M119898119899
such that
120601 (119864(119898)
119894119894otimes 119868119899) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119868119903119894
0
0 0 0119898119899minus119904minus119903
119894
]
]
]
]
]
119875minus1
1 (22)
where 119903119894= rank 120601 (119864(119898)
119894119894otimes 119868119899) and 119904
119894= 1199031+ sdot sdot sdot + 119903
119894minus1 For any
119895 isin [1 119899] it follows from 119864(119898)
119894119894otimes 119864(119899)
119895119895and 119864(119898)
119894119894otimes (119868119899minus 119864(119899)
119895119895)
being idempotent matrices in T119898otimes T119899that 120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895)
and 120601 (119864(119898)119894119894
otimes 119868119899) minus 120601 (119864
(119898)
119894119894otimes 119864(119899)
119895119895) are idempotent matrices
we obtain by (22) and Lemma 6 that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119883119895
0
0 0 0119898119899minus119904
119894
minus119903119894
]
]
]
]
]
119875minus1
1
forall119895 isin [1 119899]
(23)
where 119883119895isin M119903119894
is an idempotent matrix For any 119895 = 119897 isin
[1 119899]119864(119898)119894119894
otimes(119864(119899)
119895119895+119864(119899)
119897119897) is an idempotentmatrix inT
119898otimesT119899
we have 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119894119894otimes 119864(119899)
119897119897) is an idempotent
matrix It follows from (23) that 119883119895+ 119883119897is an idempotent
matrix If 119903119894lt 119899 by Lemma 9 we can obtain that there exists
some 1198950isin [1 119899] such that 119883
1198950
= 0 This together with(23) implies 120601 (119864(119898)
119894119894otimes 119864(119899)
1198950
1198950
) = 0 which is a contradictionto the fact that 120601 is injective Hence 119903
119894= 119899 forall119894 isin [1 119899] By
Lemma 9 there exists an invertible matrix119876119894isin M119899such that
119883119895= 119876119894119864(119899)
119895119895119876minus1
119894 forall119895 isin [1 119899] Let 119876 = diag (119876
1 119876
119898) it
follows from (23) that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751119876(119864(119898)
119894119894otimes 119864(119899)
119895119895)119876minus1
119875minus1
1
forall119894 isin [1119898] 119896 isin [1 119899]
(24)
Hence (20) and (21) hold This completes the proof of Step 1By Step 1 we may assume that for any 119894 isin [1 119898] 119895 isin
[1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898119899)
120587(120588(119894119895))120587(120588(119894119895)) (25)
From this together with (3) we can also write
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864
(119898)
120590(120587(120588(119894119895)))120590(120587(120588(119894119895)))
otimes 119864(119899)
120591(120587(120588(119894119895)))120591(120587(120588(119894119895)))
(26)
Step 2(i) For any 119894 isin [1 119898] 1198951 1198952
isin [1 119899] (120590(120587(120588(1198941198951))) 120591(120587(120588(119894 119895
1)))) and (120590(120587(120588(119894 119895
2))) 120591(120587(120588(119894 119895
2)))) are
comparable(ii) For any 119894
1 1198942isin [1119898] 119895 isin [1 119899] (120590(120587(120588(119894
1 119895)))
120591(120587(120588(1198941 119895)))) and (120590(120587(120588(119894
2 119895))) 120591(120587(120588(119894
2 119895)))) are compa-
rableProof of Step 2 (i) Suppose there exist some 119894
0isin [1119898] and
1198951lt 1198952isin [1 119899] such that (120590(120587(120588(119894
0 1198951))) 120591(120587(120588(119894
0 1198951)))) and
(120590(120587(120588(1198940 1198952))) 120591(120587(120588(119894
0 1198952)))) are not comparable Without
loss of generality we may assume that
120590 (120587 (120588 (1198940 1198951))) lt 120590 (120587 (120588 (119894
0 1198952)))
120591 (120587 (120588 (1198940 1198951))) gt 120591 (120587 (120588 (119894
0 1198952)))
(27)
It follows that
120587 (120588 (1198940 1198951)) = (120590 (120587 (120588 (119894
0 1198951))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198951)))
le (120590 (120587 (120588 (1198940 1198951))) minus 1) 119899 + 119899
le (120590 (120587 (120588 (1198940 1198951)))) 119899
le (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899
lt (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198952)))
= 120587 (120588 (1198940 1198952))
(28)
Abstract and Applied Analysis 5
For any 119909 isin C by 119864(119898)1198940
1198940
otimes(119864(119899)
1198951
1198951
+119909119864(119899)
1198951
1198952
) and 119864(119898)1198940
1198940
otimes(119864(119899)
1198952
1198952
+
119909119864(119899)
1198951
1198952
) being idempotent matrices inT119898otimesT119899 we obtain by
(25) that
119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198951
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
119864(119898119899)
120587(120588(1198940
1198952
))120587(120588(1198940
1198952
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
(29)
are idempotent matrices inT119898otimesT119899 hence by Lemma 8
120601 (119864(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
) = 120582119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198952
)) (30)
From (27) by Lemma 4 we obtain that 120582 = 0 which is acontradiction to the fact that 120601 is injective Using a similarmethod we may prove (ii) holds This completes the proof ofStep 2Note It is easy to see that 120588 is a bijective map from [1 119898] times
[1 119899] to [1 119898119899] with 120588minus1
119896 997891rarr (120590(119896) 120591(119896)) from [1 119898119899]
to [1 119898] times [1 119899] This together with 120587 is a permutation on[1 119898119899] we obtain that
(120590 (120587120588 (119894 119895)) 120591 (120587120588 (119894 119895))) 119894 isin [1 119898] 119895 isin [1 119899]
= [1119898] times [1 119899]
(31)
Let 119886119894119895= (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) then [119886
119894119895] forms an 119898 times 119899
matrix containing all elements of [1 119898]times[1 119899] Step 2 impliesthat every two elements of [119886
119894119895] in the same row and column
are comparable respectivelyThus applying Lemma 5 to [119886119894119895]
we conclude that one of (I)ndash(IV) holds If (I) holds thenfor any but fixed 119894 isin [1 119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895)))119895 = 1 119899 are in the same row that is 120590(120587(120588(119894 119895))) =
120590(120587(120588(119894 1))) forall119895 isin [1 119899] and 120591(120587120588(119894 119895)) 119895 = 1 119899 =
[1 119899] hence it follows from (26) that
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
120590(120587(120588(1198941)))120590(120587(120588(1198941)))otimes 119868119899 (32)
Similarly if (III) holds then 119898 = 119899 and for any but fixed119895 isin [1119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) 119894 = 1 119898 are in thesame row that is 120590(120587120588(119894 119895)) = 120590(120587120588(1 119895)) forall119894 isin [1119898] and120591(120587120588(119894 119895)) 119894 = 1 119898 = [1119898] hence it follows from(26) that
120601 (119868119898otimes 119864(119898)
119895119895) = 119864(119898)
120590(120587(120588(1119895)))120590(120587(120588(1119895)))otimes 119868119898 (33)
We claim that (II) and (IV) do not hold Indeed if (II)holds for convenience we assume 119898 ge 3 and we firstconsider the special case 119880 = 119868
119898 119881 = 119868
119899in (II) (one can
use a similar method to prove the case of 119899 ge 3) Thus by(26) we have
120601 (119864(119898)
11otimes 119864(119898)
11) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119898119898otimes 119864(119899)
119899119899) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(34)
Since for any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
11+
119909119864(119899)
12) and (119864
(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
22+ 119909119864(119899)
12) are
idempotent matrices inT119898otimesT119899 we obtain by (34) that
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
119864(119898)
22otimes 119864(119899)
11+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
11otimes 119864(119899)
22+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
(35)
are idempotent matrices in T119898otimes T119899 This together with
Lemma 8 implies
120601 (119864(119898)
12otimes 119864(119899)
11) = 120582119864
(119898119899)
(119899+1)119898119899= 0
120601 (119864(119898)
11otimes 119864(119899)
12) = 120583119864
(119898119899)
2119898119899= 0
(36)
For any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes (119864
(119899)
11+119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 Thus by (34) and (36)
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ119896120573119896119864(119898119899)
119896119898119899 (37)
Similarly since for any 119909 isin C (119864(119898)22
+ 119909119864(119898)
12) otimes 119864(119899)
22 (119864(119898)11
+
119909119864(119898)
12) otimes 119864(119899)
22and 119864(119898)
22otimes (119864(119899)
22+ 119909119864(119899)
12) 119864(119898)22
otimes (119864(119899)
11+ 119909119864(119899)
12)
are idempotent matrices inT119898otimesT119899 we have
120601 (119864(119898)
12otimes 119864(119899)
22) = 1205821015840
119864(119898119899)
2(119899+2)= 0
120601 (119864(119898)
22otimes 119864(119899)
12) = 1205831015840
119864(119898119899)
(119899+1)(119899+2)= 0
(38)
For any 119909 isin C (119864(119898)22
+ 119864(119898)
12) otimes (119864
(119899)
22+ 119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 we obtain
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ1198961205731015840
119896119864(119898119899)
119896(119899+2) (39)
It follows from (37) and (39) that 120601 (119864(119898)12
otimes119864(119899)
12) = 0 which is
a contradiction to that 120601 is injectiveFor general case by (II) we can choose a permutation
1199011 119901
119898of [1119898] and a permutation 119902
1 119902
119899of [1 119899] such
that
(120590 (120587 (120588 (1199011 1199021))) 120591 (120587 (120588 (119901
1 1199021)))) = (119898 119899)
(120590 (120587 (120588 (119901119898 119902119899))) 120591 (120587 (120588 (119901
119898 119902119899)))) = (1 1)
(120590 (120587 (120588 (119901119894 119902119895))) 120591 (120587 (120588 (119901
119894 119902119895)))) = (119894 119895)
forall (119894 119895) = (1 1) (119898 119899)
(40)
From this together with (26) we obtain
120601 (119864(119898)
1199011
1199011
otimes 119864(119898)
1199021
1199021
) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119901119898
119901119898
otimes 119864(119899)
119902119899
119902119899
) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119901119894
119901119894
otimes 119864(119899)
119902119895
119902119895
) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(41)
6 Abstract and Applied Analysis
Using a similar method as the above we can drive a contra-diction Similarly we may prove (IV) does not hold
If (32) holds we may assume
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
119892(119894)119892(119894)otimes 119868119899 (42)
where 119892 is a permutation on [1 119898] If (33) holds we mayassume
120601 (119868119898otimes 119864(119898)
119894119894) = 119864(119898)
119892(119894)119892(119894)otimes 119868119898 (43)
We next assume (42) to prove (i) of theorem holds and onecan use similarmethods to prove (ii) of theorem if (43) holdsStep 3 There exists an invertible matrix 119875 isin T
119898otimesT119899such
that
120601 (119864(119898)
119894119894otimes 119883) = 119875 (119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883)) 119875
minus1
forall119894 isin [1119898] 119883 isin T119899
(44)
where for 119894 isin [1119898] 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869Proof of Step 3 For any idempotent matrix 119860 isin T
119899 since
119864(119898)
119894119894otimes119860 and 119864(119898)
119894119894otimes(119868119899minus119860) are idempotent matrices inT
119898otimes
T119899 we obtain by (42) that
120601 (119864(119898)
119894119894otimes 119860) 119864
(119898)
119892(119894)119892(119894)otimes 119868119899minus 120601 (119864
(119898)
119894119894otimes 119860) (45)
are idempotent matrices inT119898otimesT119899 By Lemma 7 we have
120601 (119864(119898)
119894119894otimes 119860) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119860) (46)
where 120595119894(119860) isin T
119899is an idempotent matrix By the
arbitrariness of119860 we can expand120595119894to be a linearmap onT
119899
Hence
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119883) forall119883 isin T
119899 (47)
It is easy to see that120595119894is injective and preserving idempotents
Thus by Lemma 3 there exists an invertible 119875119892(119894)
isin T119899such
that 120595119894(119883) = 119875
119892(119894)120585119894(119883)119875minus1
119892(119894) where 120585
119894(119883) = 119883 or 120585
119894(119883) =
119869119883119905
119869 Let 119875 = diag (1198751 119875
119898) isin T
119898otimesT119899 we complete the
proof of this stepBy Step 3 we may assume
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883) forall119894 isin [1119898] 119883 isin T
119899
(48)
Step 4 119892 (119894) = 119894 or 119892(119894) = 119898 minus 119894 + 1Proof of Step 4 If 119898 = 2 then this claim is clear For 119898 ge 3we prove that if 119894 lt 119895 lt 119896 then 119892 (119894) lt 119892 (119895) lt 119892 (119896) or 119892 (119894) gt119892 (119895) gt 119892 (119896) Otherwise we assume 119892 (119894) lt 119892 (119896) lt 119892 (119895)
(other cases can be proven by using similar methods) Sincefor any 119909 isin C (119864(119898)
119894119894+ 119909119864(119898)
119894119895) otimes 119868119899and (119864(119898)
119895119895+ 119909119864(119898)
119894119895) otimes 119868119899are
idempotent matrices inT119898otimesT119899 we have by using (48) that
119864(119898)
119892(119894)119892(119894)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899) and 119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899)
are idempotent matrices in 119879119898otimes 119879119899 This together with
Lemma 8 implies
120601 (119864(119898)
119894119895otimes 119868119899) = 119864(119898)
119892(119894)119892(119895)otimes 119860 for some119860 = 0 isin T
119899 (49)
Similarly
120601 (119864(119898)
119894119896otimes 119868119899) = 119864(119898)
119892(119894)119892(119896)otimes 119861 for some119861 = 0 isin T
119899
120601 (119864(119898)
119895119896otimes 119868119899) = 119864(119898)
119892(119896)119892(119895)otimes 119862 for some119862 = 0 isin T
119899
(50)
By (119864(119898)119895119895
+119864(119898)
119894119895+119864(119898)
119894119896+119864(119898)
119895119896)otimes119868119899being an idempotent matrix
inT119898otimesT119899 we obtain by using (48) (49) and (50) that
119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119864(119898)
119892(119894)119892(119895)otimes 119860 + 119864
(119898)
119892(119894)119892(119896)otimes 119861 + 119864
(119898)
119892(119896)119892(119895)otimes 119862
(51)
is an idempotent matrix inT119898otimesT119899 that is
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
2
=
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
(52)
This implies 119860 = 0 which is a contradiction Hence wecomplete the proof of Step 4Step 5 For any 119894 lt 119895 120585
119894= 120585119895 and there exists 120582
119894119895= 0 such that
120601 (119864(119898)
119894119895otimes 119883) = 120582
119894119895119864(119898)
119892(119894)119892(119895)otimes 120585119894(119883) forall119894 = 119895 119883 isin T
119899
(53)
Proof of Step 5 We prove the case of 119892 (119894) = 119894 (one can use asimilar method to prove the case of 119892 (119894) = 119898 minus 119894 + 1) Hence
120601 (119864(119898)
119894119894otimes 119864(119899)
119896119896) = 119864(119898)
119894119894otimes 119864(119899)
119896119896 forall119894 isin [1119898] 119896 isin [1 119899]
(54)
Without loss of generality we may assume 119894 = 1 119895 = 2 Sincefor any 119909 isin C (119864(119898)
11+ 119909119864(119898)
12) otimes 119864(119899)
119896119896and (119864(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
119896119896
are idempotent matrices inT119898otimesT119899 we obtain by (54) that
119864(119898)
11otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) and119864(119898)
22otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) are
idempotentmatrices in119879119898otimes119879119899This together with Lemma 8
implies 120601(119864(119898)12
otimes 119864(119899)
119896119896) = 120582119896119864(119898)
12otimes 119864(119899)
119896119896 where 120582
119896= 0 Let Λ =
diag (1205821 120582
119898) then
120601 (119864(119898)
12otimes 119868119899) =
[
[
0 Λ
0 0
]
]
oplus 0 (55)
Let119876 = [119868119898
minusΛ
0 119868119898
]oplus119868(119898minus2)119899
isin T119898otimesT119899 by (55) one can obtain
120601 ((119864(119898)
11+ 119864(119898)
12) otimes 119868119899) =
[
[
119868119899Λ
0 0
]
]
oplus 0
= 119876 (119864(119898)
11otimes 119868119899)119876minus1
120601 ((119864(119898)
22minus 119864(119898)
12) otimes 119868119899) =
[
[
0 minusΛ
0 119868119899
]
]
oplus 0
= 119876 (119864(119898)
22otimes 119868119899)119876minus1
(56)
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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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
2 Abstract and Applied Analysis
forall119894 119895 isin [1 119899] stands for the 119899 times 119899 matrix with 1 at the(119894 119895)th entry and 0 otherwise Denote by 119869
119896(also 119869) thematrix
119864(119896)
1119896+ 119864(119896)
2119896minus1+ sdot sdot sdot + 119864
(119896)
1198961 Clearly if 119860 = [119886
119894119895] isin T
119899 then
119869119860119905
119869 = [119886119899minus119895+1119899minus119894+1
] isin T119899 For positive integers 119899
1and 119899
2
with 1198991lt 1198992 let [119899
1 1198992] be the set of all integers between 119899
1
and 1198992 For any (119894 119895) isin [1119898] times [1 119899] we define 120588 by
120588 (119894 119895) = (119894 minus 1) 119899 + 119895 (1)
For any 119896 isin [1119898119899] we define 120590(119896) isin [1119898] and 120591 (119896) isin [1 119899]such that 119896 = (120590(119896) minus 1)119899 + 120591(119896) (it is easy to see that 120590 and 120591are well defined) It is easy to see that
119864(119898)
119903119904otimes 119864(119899)
119906V = 119864(119898119899)
(119903minus1)119899+119906(119904minus1)119899+V = 119864(119898119899)
120588(119903119906)120588(119904V) (2)
119864(119898119899)
119894119895= 119864(119898)
120590(119894)120590(119895)otimes 119864(119899)
120591(119894)120591(119895) (3)
We define a partial ordering of [1 119898]times[1 119899] by (119886 119887) le (119888 119889)
if and only if 119886 le 119888 and 119887 le 119889 We say that (119886 119887) and (119888 119889) arecomparable if (119886 119887) le (119888 119889) or (119888 119889) le (119886 119887)
2 Preliminary Results
We need the form of injective linear idempotence preserveronT119899 which was obtained in [5]
Lemma 3 (see [5 Theorem 1]) Let 120595 be an injective linearmap on T
119899 Then 120595 (119883) is an idempotent matrix in T
119899
whenever119883 is an idempotent matrix inT119899if and only if there
exists an invertible matrix 119875 isin T119899such that
120595 (119883) = 119875120585 (119883) 119875minus1
forall119883 isin T119899 (4)
where 120585(119883) = 119883 or 120585(119883) = 119869119883119905
119869
It is clear thatT119898otimesT119899⊊ T119898119899 For example 119864(4)
23isin T4
and
119864(4)
23notin T2otimesT2=
[
[
[
[
[
[
[
[
[
11988611
11988612
11988613
11988614
0 11988622
0 11988624
0 0 11988633
11988634
0 0 0 11988644
]
]
]
]
]
]
]
]
]
119886119894119895isin C
(5)
It is easy to see that (120590(2) 120591(2)) = (1 2) and (120590(3) 120591(3)) =
(2 1) In fact we can point out the positions of elementswhich are inT
119898119899T119898otimesT119899
Lemma 4 120582119864(119898119899)119894119895
isin T119898otimes T119899if and only if (120590(119894) 120591(119894)) le
(120590(119895) 120591(119895)) or 120582 = 0
Proof It is a direct corollary of (3)
The next Lemma describes the partial ordering wedefined in Section 1 which is useful to prove our mainTheorem
Lemma 5 (see [19 Theorem 1]) Let 119898 119899 ge 2 and let 119860 bea matrix with 119898 rows and 119899 columns containing all elementsof [1 119898] times [1 119899] If every two elements of 119860 in the samerow and column are comparable respectively then there existpermutation matrices 119880 isin M
119898and 119881 isin M
119899such that
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
[
[
(1 1) (1 2) sdot sdot sdot (1 119899 minus 1) (1 119899)
(2 1) (2 2) sdot sdot sdot (2 119899 minus 1) (2 119899)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(119898 minus 1 1) (119898 minus 1 2) sdot sdot sdot (119898 minus 1 119899 minus 1) (119898 minus 1 119899)
(119898 1) (119898 2) sdot sdot sdot (119898 119899 minus 1) (119898 119899)
]
]
]
]
]
]
]
]
]
]
]
]
]
(I)
or when119898 ge 3 or 119899 ge 3
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
(119898 119899) (1 2) sdot sdot sdot (1 119899 minus 1) (1 119899)
(2 1) (2 2) sdot sdot sdot (2 119899 minus 1) (2 119899)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(119898 minus 1 1) (119898 minus 1 2) sdot sdot sdot (119898 minus 1 119899 minus 1) (119898 minus 1 119899)
(119898 1) (119898 2) sdot sdot sdot (119898 119899 minus 1) (1 1)
]
]
]
]
]
]
]
]
]
]
]
(II)
or when119898 = 119899
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
(1 1) (2 1) sdot sdot sdot (119898 minus 1 1) (119898 1)
(1 2) (2 2) sdot sdot sdot (119898 minus 1 2) (119898 2)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(1 119898 minus 1) (2119898 minus 1) sdot sdot sdot (119898 minus 1119898 minus 1) (119898119898 minus 1)
(1119898) (2119898) sdot sdot sdot (119898 minus 1119898) (119898119898)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(III)
or when119898 = 119899 ge 3
119880119860119881
=
[
[
[
[
[
[
[
[
[
[
[
[
(119898119898) (2 1) sdot sdot sdot (119898 minus 1 1) (119898 1)
(1 2) (2 2) sdot sdot sdot (119898 minus 1 2) (119898 2)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
(1 119898 minus 1) (2119898 minus 1) sdot sdot sdot (119898 minus 1119898 minus 1) (119898119898 minus 1)
(1119898) (2119898) sdot sdot sdot (119898 minus 1119898) (1 1)
]
]
]
]
]
]
]
]
]
]
]
]
(IV)
Abstract and Applied Analysis 3
The following lemmas would make the proof of the maintheorem more concise
Lemma6 Suppose119883 isin M119899is an idempotentmatrix such that
0119904oplus119868119903oplus0119899minus119903minus119904
minus119883 also is an idempotentmatrixThen there existsan idempotent matrix119883
119903isin M119903such that119883 = 0
119904oplus119883119903oplus0119899minus119903minus119904
Proof By 1198832 = 119883 and (0119904oplus 119868119903oplus 0119899minus119903minus119904
minus 119883)2
= 0119904oplus 119868119903oplus
0119899minus119903minus119904
minus 119883 we have
2119883 = (0119904oplus 119868119903oplus 0119899minus119903minus119904
)119883 + 119883 (0119904oplus 119868119903oplus 0119899minus119903minus119904
) (6)
Set
119883 =
[
[
[
[
[
11988311
11988312
11988313
11988321
11988322
11988323
11988331
11988332
11988333
]
]
]
]
]
(7)
where11988311isin M119904 11988322isin M119903 Then (6) implies
2
[
[
[
[
[
11988311
11988312
11988313
11988321
11988322
11988323
11988331
11988332
11988333
]
]
]
]
]
=
[
[
[
[
[
0 11988312
0
11988321
211988322
11988323
0 11988332
0
]
]
]
]
]
(8)
Hence 119883 = 0119904oplus 11988322
oplus 0119899minus119903minus119904
therefore the lemma holds
Lemma 7 Let 119883 isin T119898otimesT119899be an idempotent matrix such
that 119864(119898)119894119894
otimes 119868119899minus 119883 also is an idempotent matrix Then there
exists an idempotent 119884 isin T119899such that 119883 = 119864
(119898)
119894119894otimes 119884
Proof The proof is similar to that of Lemma 6
Lemma 8 Suppose 119903 119904 isin [1 119899 minus 1] If for any 120582 isin C 119868119903oplus
0119899minus119903
+120582119883 and 0119903oplus119868119904oplus0119899minus119903minus119904
+120582119883 are idempotent inT119899 then
X = [
01199031198831
0 0119904
] oplus 0119899minus119903minus119904
(9)
Proof It follows from 119868119903oplus 0119899minus119903
+ 120582119883 forall120582 isin C is idempotentthat
(119868119903oplus 0119899minus119903
)119883 + 119883 (119868119903oplus 0119899minus119903
) = 119883 (10)
Let
119883 =[
[
[
1198831199031198831
1198832
0 119883119904
1198833
0 0 119883119899minus119903minus119904
]
]
]
(11)
where119883119903isin T119903119883119904isin T119904 then (10) implies
[
[
[
[
[
211988311990311988311198832
0 0 0
0 0 0
]
]
]
]
]
=
[
[
[
[
[
1198831199031198831
1198832
0 119883119904
1198833
0 0 119883119899minus119903minus119904
]
]
]
]
]
(12)
Hence 119883119903= 0 119883
119904= 0 119883
119899minus119903minus119904= 0 119883
3= 0 Similarly from
0119903oplus 119868119904oplus 0119899minus119903minus119904
+ 120582119883 being idempotent we have1198832= 0
Lemma 9 (see [6 Page 62 Exercise 1]) Suppose1198601 119860
119896isin
M119899are idempotent matrices such that for any 119894 = 119895 isin [1 119896]
119860119894+ 119860119895is idempotent Let 119903
119894= rank119860
119894 Then there exists an
invertible matrix 119875 isin M119899such that
119860119894= 119875 diag (0 0 1 1 0 0) 119875minus1 (13)
where diag (0 0 1 1 0 0) is the diagonalmatrix inwhich all diagonal entries are zero except those in the (119903
1+sdot sdot sdot+
119903119894minus1
+ 1)st to the (1199031+ sdot sdot sdot + 119903
119894)th rows
Similar to Lemma 9 we have the following
Lemma 10 Let 1198601 119860
119898119899isin T119898otimes T119899be idempotent
matrices of rank-1 such that for any 119894 = 119895 isin [1119898119899] 119860119894+ 119860119895is
idempotent Then there exist a permutation 120587 on [1 119898119899] andan invertible matrix 119875 isin T
119898otimesT119899such that
119860119894= 119875119864(119898119899)
120587(119894)120587(119894)119875minus1
119894 = 1 119898119899 (14)
Proof By119860119894isin T119898otimesT119899being an idempotentmatrix of rank-
1 we can assume
119860119894= 119864(119898119899)
120587(119894)120587(119894)+ 119861119894 119894 = 1 119898119899 (15)
where 119861119894isin T119898otimes T119899with zero diagonal entries It follows
from 119860119894+ 119860119895 forall119894 = 119895 being is idempotent that 120587(119894) = 120587(119895)
forall119894 = 119895 Hence 120587 is a permutation on [1 119898119899] By 1198602120587minus1
(1)=
119860120587minus1
(1) we can see 119860
120587minus1
(1)= 119864(119898119899)
11+ Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896 Let
1198751= 119868119898119899
minus Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896isin T119898otimes T119899 then 119875
minus1
1= 119868119898119899
+
Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896and
119860120587minus1
(1)= 1198751119864(119898119899)
11119875minus1
1 (16)
By 119860120587minus1
(1)+ 119860
120587minus1
(2)being idempotent we obtain
119875minus1
1119860120587minus1
(2)1198751
= 119864(119898119899)
22+ Σ(119898119899)
119896=31205822119896119864(119898119899)
2119896 Let 119875
2=
119868119898119899
minus Σ(119898119899)
119896=31205822119896119864(119898119899)
2119896 then
119860120587minus1
(1)= 11987521198751119864(119898119899)
11119875minus1
1119875minus1
2
119860120587minus1
(2)= 11987521198751119864(119898119899)
22119875minus1
1119875minus1
2
(17)
Continuing to do this we can find 1198753 119875
119898119899 Let 119875 =
119875119898119899119875119898119899minus1
sdot sdot sdot 11987521198751isin T119898otimesT119899 then
119860120587minus1
(119894)= 119875119864(119898119899)
119894119894119875minus1
119894 = 1 119898119899 (18)
This completes the proof
3 The Main Result
Themain result of this paper is as follows
Theorem 11 Suppose 119898 119899 ge 2 are positive integers and 120601
is an injective linear map on T119898otimes T119899 Then 120601 (119860 otimes 119861)
4 Abstract and Applied Analysis
is an idempotent matrix in T119898otimes T119899whenever 119860 otimes 119861 is an
idempotent matrix in T119898otimes T119899if and only if there exists an
invertible matrix 119875 isin T119898otimesT119899such that
120601 (119860 otimes 119861) = 119875 (1205851(119860) otimes 120585
2(119861)) 119875
minus1
forall119860 isin T119898 119861 isin T
119899
(i)
or when119898 = 119899
120601 (119860 otimes 119861) = 119875 (1205851(119861) otimes 120585
2(119860)) 119875
minus1
forall119860 isin T119898 119861 isin T
119898
(ii)
where for 119894 = 1 2 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869
Proof The sufficiency is obvious We will prove the necessityby the following six steps
Step 1 There exist a permutation 120587 on [1 119898119899] and aninvertible matrix 119875 isin T
119898otimesT119899such that
120601 (119864(119898119899)
119896119896) = 119875119864
(119898119899)
120587(119896)120587(119896)119875minus1
forall119896 isin [1119898119899] (19)
Proof of Step 1 By Lemma 10 we only need to prove that
rank 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) = 1 forall119894 isin [1119898] 119895 isin [1 119899] (20)
And for any (119894 119895) = (119906 V) isin [1119898] times [1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119906119906otimes 119864(119899)
VV ) is an idempotentmatrix(21)
It follows from 119864(119898)
11otimes 119868119899 119864
(119898)
119898119898otimes 119868119899and (119864(119898)
119894119894+119864(119898)
119895119895) otimes
119868119899forall119894 = 119895 isin [1119898] are idempotent matrices in T
119898otimes T119899that
120601 (119864(119898)
11otimes119868119899) 120601 (119864
(119898)
119898119898otimes119868119899) and 120601 (119864(119898)
119894119894otimes119868119899)+120601 (119864
(119898)
119895119895otimes119868119899)
forall119894 = 119895 isin [1119898] are idempotent matrices We obtain by usingLemma 9 that there exists an invertible matrix 119875
1isin M119898119899
such that
120601 (119864(119898)
119894119894otimes 119868119899) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119868119903119894
0
0 0 0119898119899minus119904minus119903
119894
]
]
]
]
]
119875minus1
1 (22)
where 119903119894= rank 120601 (119864(119898)
119894119894otimes 119868119899) and 119904
119894= 1199031+ sdot sdot sdot + 119903
119894minus1 For any
119895 isin [1 119899] it follows from 119864(119898)
119894119894otimes 119864(119899)
119895119895and 119864(119898)
119894119894otimes (119868119899minus 119864(119899)
119895119895)
being idempotent matrices in T119898otimes T119899that 120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895)
and 120601 (119864(119898)119894119894
otimes 119868119899) minus 120601 (119864
(119898)
119894119894otimes 119864(119899)
119895119895) are idempotent matrices
we obtain by (22) and Lemma 6 that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119883119895
0
0 0 0119898119899minus119904
119894
minus119903119894
]
]
]
]
]
119875minus1
1
forall119895 isin [1 119899]
(23)
where 119883119895isin M119903119894
is an idempotent matrix For any 119895 = 119897 isin
[1 119899]119864(119898)119894119894
otimes(119864(119899)
119895119895+119864(119899)
119897119897) is an idempotentmatrix inT
119898otimesT119899
we have 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119894119894otimes 119864(119899)
119897119897) is an idempotent
matrix It follows from (23) that 119883119895+ 119883119897is an idempotent
matrix If 119903119894lt 119899 by Lemma 9 we can obtain that there exists
some 1198950isin [1 119899] such that 119883
1198950
= 0 This together with(23) implies 120601 (119864(119898)
119894119894otimes 119864(119899)
1198950
1198950
) = 0 which is a contradictionto the fact that 120601 is injective Hence 119903
119894= 119899 forall119894 isin [1 119899] By
Lemma 9 there exists an invertible matrix119876119894isin M119899such that
119883119895= 119876119894119864(119899)
119895119895119876minus1
119894 forall119895 isin [1 119899] Let 119876 = diag (119876
1 119876
119898) it
follows from (23) that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751119876(119864(119898)
119894119894otimes 119864(119899)
119895119895)119876minus1
119875minus1
1
forall119894 isin [1119898] 119896 isin [1 119899]
(24)
Hence (20) and (21) hold This completes the proof of Step 1By Step 1 we may assume that for any 119894 isin [1 119898] 119895 isin
[1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898119899)
120587(120588(119894119895))120587(120588(119894119895)) (25)
From this together with (3) we can also write
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864
(119898)
120590(120587(120588(119894119895)))120590(120587(120588(119894119895)))
otimes 119864(119899)
120591(120587(120588(119894119895)))120591(120587(120588(119894119895)))
(26)
Step 2(i) For any 119894 isin [1 119898] 1198951 1198952
isin [1 119899] (120590(120587(120588(1198941198951))) 120591(120587(120588(119894 119895
1)))) and (120590(120587(120588(119894 119895
2))) 120591(120587(120588(119894 119895
2)))) are
comparable(ii) For any 119894
1 1198942isin [1119898] 119895 isin [1 119899] (120590(120587(120588(119894
1 119895)))
120591(120587(120588(1198941 119895)))) and (120590(120587(120588(119894
2 119895))) 120591(120587(120588(119894
2 119895)))) are compa-
rableProof of Step 2 (i) Suppose there exist some 119894
0isin [1119898] and
1198951lt 1198952isin [1 119899] such that (120590(120587(120588(119894
0 1198951))) 120591(120587(120588(119894
0 1198951)))) and
(120590(120587(120588(1198940 1198952))) 120591(120587(120588(119894
0 1198952)))) are not comparable Without
loss of generality we may assume that
120590 (120587 (120588 (1198940 1198951))) lt 120590 (120587 (120588 (119894
0 1198952)))
120591 (120587 (120588 (1198940 1198951))) gt 120591 (120587 (120588 (119894
0 1198952)))
(27)
It follows that
120587 (120588 (1198940 1198951)) = (120590 (120587 (120588 (119894
0 1198951))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198951)))
le (120590 (120587 (120588 (1198940 1198951))) minus 1) 119899 + 119899
le (120590 (120587 (120588 (1198940 1198951)))) 119899
le (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899
lt (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198952)))
= 120587 (120588 (1198940 1198952))
(28)
Abstract and Applied Analysis 5
For any 119909 isin C by 119864(119898)1198940
1198940
otimes(119864(119899)
1198951
1198951
+119909119864(119899)
1198951
1198952
) and 119864(119898)1198940
1198940
otimes(119864(119899)
1198952
1198952
+
119909119864(119899)
1198951
1198952
) being idempotent matrices inT119898otimesT119899 we obtain by
(25) that
119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198951
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
119864(119898119899)
120587(120588(1198940
1198952
))120587(120588(1198940
1198952
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
(29)
are idempotent matrices inT119898otimesT119899 hence by Lemma 8
120601 (119864(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
) = 120582119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198952
)) (30)
From (27) by Lemma 4 we obtain that 120582 = 0 which is acontradiction to the fact that 120601 is injective Using a similarmethod we may prove (ii) holds This completes the proof ofStep 2Note It is easy to see that 120588 is a bijective map from [1 119898] times
[1 119899] to [1 119898119899] with 120588minus1
119896 997891rarr (120590(119896) 120591(119896)) from [1 119898119899]
to [1 119898] times [1 119899] This together with 120587 is a permutation on[1 119898119899] we obtain that
(120590 (120587120588 (119894 119895)) 120591 (120587120588 (119894 119895))) 119894 isin [1 119898] 119895 isin [1 119899]
= [1119898] times [1 119899]
(31)
Let 119886119894119895= (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) then [119886
119894119895] forms an 119898 times 119899
matrix containing all elements of [1 119898]times[1 119899] Step 2 impliesthat every two elements of [119886
119894119895] in the same row and column
are comparable respectivelyThus applying Lemma 5 to [119886119894119895]
we conclude that one of (I)ndash(IV) holds If (I) holds thenfor any but fixed 119894 isin [1 119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895)))119895 = 1 119899 are in the same row that is 120590(120587(120588(119894 119895))) =
120590(120587(120588(119894 1))) forall119895 isin [1 119899] and 120591(120587120588(119894 119895)) 119895 = 1 119899 =
[1 119899] hence it follows from (26) that
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
120590(120587(120588(1198941)))120590(120587(120588(1198941)))otimes 119868119899 (32)
Similarly if (III) holds then 119898 = 119899 and for any but fixed119895 isin [1119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) 119894 = 1 119898 are in thesame row that is 120590(120587120588(119894 119895)) = 120590(120587120588(1 119895)) forall119894 isin [1119898] and120591(120587120588(119894 119895)) 119894 = 1 119898 = [1119898] hence it follows from(26) that
120601 (119868119898otimes 119864(119898)
119895119895) = 119864(119898)
120590(120587(120588(1119895)))120590(120587(120588(1119895)))otimes 119868119898 (33)
We claim that (II) and (IV) do not hold Indeed if (II)holds for convenience we assume 119898 ge 3 and we firstconsider the special case 119880 = 119868
119898 119881 = 119868
119899in (II) (one can
use a similar method to prove the case of 119899 ge 3) Thus by(26) we have
120601 (119864(119898)
11otimes 119864(119898)
11) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119898119898otimes 119864(119899)
119899119899) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(34)
Since for any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
11+
119909119864(119899)
12) and (119864
(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
22+ 119909119864(119899)
12) are
idempotent matrices inT119898otimesT119899 we obtain by (34) that
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
119864(119898)
22otimes 119864(119899)
11+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
11otimes 119864(119899)
22+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
(35)
are idempotent matrices in T119898otimes T119899 This together with
Lemma 8 implies
120601 (119864(119898)
12otimes 119864(119899)
11) = 120582119864
(119898119899)
(119899+1)119898119899= 0
120601 (119864(119898)
11otimes 119864(119899)
12) = 120583119864
(119898119899)
2119898119899= 0
(36)
For any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes (119864
(119899)
11+119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 Thus by (34) and (36)
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ119896120573119896119864(119898119899)
119896119898119899 (37)
Similarly since for any 119909 isin C (119864(119898)22
+ 119909119864(119898)
12) otimes 119864(119899)
22 (119864(119898)11
+
119909119864(119898)
12) otimes 119864(119899)
22and 119864(119898)
22otimes (119864(119899)
22+ 119909119864(119899)
12) 119864(119898)22
otimes (119864(119899)
11+ 119909119864(119899)
12)
are idempotent matrices inT119898otimesT119899 we have
120601 (119864(119898)
12otimes 119864(119899)
22) = 1205821015840
119864(119898119899)
2(119899+2)= 0
120601 (119864(119898)
22otimes 119864(119899)
12) = 1205831015840
119864(119898119899)
(119899+1)(119899+2)= 0
(38)
For any 119909 isin C (119864(119898)22
+ 119864(119898)
12) otimes (119864
(119899)
22+ 119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 we obtain
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ1198961205731015840
119896119864(119898119899)
119896(119899+2) (39)
It follows from (37) and (39) that 120601 (119864(119898)12
otimes119864(119899)
12) = 0 which is
a contradiction to that 120601 is injectiveFor general case by (II) we can choose a permutation
1199011 119901
119898of [1119898] and a permutation 119902
1 119902
119899of [1 119899] such
that
(120590 (120587 (120588 (1199011 1199021))) 120591 (120587 (120588 (119901
1 1199021)))) = (119898 119899)
(120590 (120587 (120588 (119901119898 119902119899))) 120591 (120587 (120588 (119901
119898 119902119899)))) = (1 1)
(120590 (120587 (120588 (119901119894 119902119895))) 120591 (120587 (120588 (119901
119894 119902119895)))) = (119894 119895)
forall (119894 119895) = (1 1) (119898 119899)
(40)
From this together with (26) we obtain
120601 (119864(119898)
1199011
1199011
otimes 119864(119898)
1199021
1199021
) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119901119898
119901119898
otimes 119864(119899)
119902119899
119902119899
) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119901119894
119901119894
otimes 119864(119899)
119902119895
119902119895
) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(41)
6 Abstract and Applied Analysis
Using a similar method as the above we can drive a contra-diction Similarly we may prove (IV) does not hold
If (32) holds we may assume
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
119892(119894)119892(119894)otimes 119868119899 (42)
where 119892 is a permutation on [1 119898] If (33) holds we mayassume
120601 (119868119898otimes 119864(119898)
119894119894) = 119864(119898)
119892(119894)119892(119894)otimes 119868119898 (43)
We next assume (42) to prove (i) of theorem holds and onecan use similarmethods to prove (ii) of theorem if (43) holdsStep 3 There exists an invertible matrix 119875 isin T
119898otimesT119899such
that
120601 (119864(119898)
119894119894otimes 119883) = 119875 (119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883)) 119875
minus1
forall119894 isin [1119898] 119883 isin T119899
(44)
where for 119894 isin [1119898] 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869Proof of Step 3 For any idempotent matrix 119860 isin T
119899 since
119864(119898)
119894119894otimes119860 and 119864(119898)
119894119894otimes(119868119899minus119860) are idempotent matrices inT
119898otimes
T119899 we obtain by (42) that
120601 (119864(119898)
119894119894otimes 119860) 119864
(119898)
119892(119894)119892(119894)otimes 119868119899minus 120601 (119864
(119898)
119894119894otimes 119860) (45)
are idempotent matrices inT119898otimesT119899 By Lemma 7 we have
120601 (119864(119898)
119894119894otimes 119860) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119860) (46)
where 120595119894(119860) isin T
119899is an idempotent matrix By the
arbitrariness of119860 we can expand120595119894to be a linearmap onT
119899
Hence
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119883) forall119883 isin T
119899 (47)
It is easy to see that120595119894is injective and preserving idempotents
Thus by Lemma 3 there exists an invertible 119875119892(119894)
isin T119899such
that 120595119894(119883) = 119875
119892(119894)120585119894(119883)119875minus1
119892(119894) where 120585
119894(119883) = 119883 or 120585
119894(119883) =
119869119883119905
119869 Let 119875 = diag (1198751 119875
119898) isin T
119898otimesT119899 we complete the
proof of this stepBy Step 3 we may assume
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883) forall119894 isin [1119898] 119883 isin T
119899
(48)
Step 4 119892 (119894) = 119894 or 119892(119894) = 119898 minus 119894 + 1Proof of Step 4 If 119898 = 2 then this claim is clear For 119898 ge 3we prove that if 119894 lt 119895 lt 119896 then 119892 (119894) lt 119892 (119895) lt 119892 (119896) or 119892 (119894) gt119892 (119895) gt 119892 (119896) Otherwise we assume 119892 (119894) lt 119892 (119896) lt 119892 (119895)
(other cases can be proven by using similar methods) Sincefor any 119909 isin C (119864(119898)
119894119894+ 119909119864(119898)
119894119895) otimes 119868119899and (119864(119898)
119895119895+ 119909119864(119898)
119894119895) otimes 119868119899are
idempotent matrices inT119898otimesT119899 we have by using (48) that
119864(119898)
119892(119894)119892(119894)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899) and 119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899)
are idempotent matrices in 119879119898otimes 119879119899 This together with
Lemma 8 implies
120601 (119864(119898)
119894119895otimes 119868119899) = 119864(119898)
119892(119894)119892(119895)otimes 119860 for some119860 = 0 isin T
119899 (49)
Similarly
120601 (119864(119898)
119894119896otimes 119868119899) = 119864(119898)
119892(119894)119892(119896)otimes 119861 for some119861 = 0 isin T
119899
120601 (119864(119898)
119895119896otimes 119868119899) = 119864(119898)
119892(119896)119892(119895)otimes 119862 for some119862 = 0 isin T
119899
(50)
By (119864(119898)119895119895
+119864(119898)
119894119895+119864(119898)
119894119896+119864(119898)
119895119896)otimes119868119899being an idempotent matrix
inT119898otimesT119899 we obtain by using (48) (49) and (50) that
119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119864(119898)
119892(119894)119892(119895)otimes 119860 + 119864
(119898)
119892(119894)119892(119896)otimes 119861 + 119864
(119898)
119892(119896)119892(119895)otimes 119862
(51)
is an idempotent matrix inT119898otimesT119899 that is
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
2
=
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
(52)
This implies 119860 = 0 which is a contradiction Hence wecomplete the proof of Step 4Step 5 For any 119894 lt 119895 120585
119894= 120585119895 and there exists 120582
119894119895= 0 such that
120601 (119864(119898)
119894119895otimes 119883) = 120582
119894119895119864(119898)
119892(119894)119892(119895)otimes 120585119894(119883) forall119894 = 119895 119883 isin T
119899
(53)
Proof of Step 5 We prove the case of 119892 (119894) = 119894 (one can use asimilar method to prove the case of 119892 (119894) = 119898 minus 119894 + 1) Hence
120601 (119864(119898)
119894119894otimes 119864(119899)
119896119896) = 119864(119898)
119894119894otimes 119864(119899)
119896119896 forall119894 isin [1119898] 119896 isin [1 119899]
(54)
Without loss of generality we may assume 119894 = 1 119895 = 2 Sincefor any 119909 isin C (119864(119898)
11+ 119909119864(119898)
12) otimes 119864(119899)
119896119896and (119864(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
119896119896
are idempotent matrices inT119898otimesT119899 we obtain by (54) that
119864(119898)
11otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) and119864(119898)
22otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) are
idempotentmatrices in119879119898otimes119879119899This together with Lemma 8
implies 120601(119864(119898)12
otimes 119864(119899)
119896119896) = 120582119896119864(119898)
12otimes 119864(119899)
119896119896 where 120582
119896= 0 Let Λ =
diag (1205821 120582
119898) then
120601 (119864(119898)
12otimes 119868119899) =
[
[
0 Λ
0 0
]
]
oplus 0 (55)
Let119876 = [119868119898
minusΛ
0 119868119898
]oplus119868(119898minus2)119899
isin T119898otimesT119899 by (55) one can obtain
120601 ((119864(119898)
11+ 119864(119898)
12) otimes 119868119899) =
[
[
119868119899Λ
0 0
]
]
oplus 0
= 119876 (119864(119898)
11otimes 119868119899)119876minus1
120601 ((119864(119898)
22minus 119864(119898)
12) otimes 119868119899) =
[
[
0 minusΛ
0 119868119899
]
]
oplus 0
= 119876 (119864(119898)
22otimes 119868119899)119876minus1
(56)
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
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
Abstract and Applied Analysis 3
The following lemmas would make the proof of the maintheorem more concise
Lemma6 Suppose119883 isin M119899is an idempotentmatrix such that
0119904oplus119868119903oplus0119899minus119903minus119904
minus119883 also is an idempotentmatrixThen there existsan idempotent matrix119883
119903isin M119903such that119883 = 0
119904oplus119883119903oplus0119899minus119903minus119904
Proof By 1198832 = 119883 and (0119904oplus 119868119903oplus 0119899minus119903minus119904
minus 119883)2
= 0119904oplus 119868119903oplus
0119899minus119903minus119904
minus 119883 we have
2119883 = (0119904oplus 119868119903oplus 0119899minus119903minus119904
)119883 + 119883 (0119904oplus 119868119903oplus 0119899minus119903minus119904
) (6)
Set
119883 =
[
[
[
[
[
11988311
11988312
11988313
11988321
11988322
11988323
11988331
11988332
11988333
]
]
]
]
]
(7)
where11988311isin M119904 11988322isin M119903 Then (6) implies
2
[
[
[
[
[
11988311
11988312
11988313
11988321
11988322
11988323
11988331
11988332
11988333
]
]
]
]
]
=
[
[
[
[
[
0 11988312
0
11988321
211988322
11988323
0 11988332
0
]
]
]
]
]
(8)
Hence 119883 = 0119904oplus 11988322
oplus 0119899minus119903minus119904
therefore the lemma holds
Lemma 7 Let 119883 isin T119898otimesT119899be an idempotent matrix such
that 119864(119898)119894119894
otimes 119868119899minus 119883 also is an idempotent matrix Then there
exists an idempotent 119884 isin T119899such that 119883 = 119864
(119898)
119894119894otimes 119884
Proof The proof is similar to that of Lemma 6
Lemma 8 Suppose 119903 119904 isin [1 119899 minus 1] If for any 120582 isin C 119868119903oplus
0119899minus119903
+120582119883 and 0119903oplus119868119904oplus0119899minus119903minus119904
+120582119883 are idempotent inT119899 then
X = [
01199031198831
0 0119904
] oplus 0119899minus119903minus119904
(9)
Proof It follows from 119868119903oplus 0119899minus119903
+ 120582119883 forall120582 isin C is idempotentthat
(119868119903oplus 0119899minus119903
)119883 + 119883 (119868119903oplus 0119899minus119903
) = 119883 (10)
Let
119883 =[
[
[
1198831199031198831
1198832
0 119883119904
1198833
0 0 119883119899minus119903minus119904
]
]
]
(11)
where119883119903isin T119903119883119904isin T119904 then (10) implies
[
[
[
[
[
211988311990311988311198832
0 0 0
0 0 0
]
]
]
]
]
=
[
[
[
[
[
1198831199031198831
1198832
0 119883119904
1198833
0 0 119883119899minus119903minus119904
]
]
]
]
]
(12)
Hence 119883119903= 0 119883
119904= 0 119883
119899minus119903minus119904= 0 119883
3= 0 Similarly from
0119903oplus 119868119904oplus 0119899minus119903minus119904
+ 120582119883 being idempotent we have1198832= 0
Lemma 9 (see [6 Page 62 Exercise 1]) Suppose1198601 119860
119896isin
M119899are idempotent matrices such that for any 119894 = 119895 isin [1 119896]
119860119894+ 119860119895is idempotent Let 119903
119894= rank119860
119894 Then there exists an
invertible matrix 119875 isin M119899such that
119860119894= 119875 diag (0 0 1 1 0 0) 119875minus1 (13)
where diag (0 0 1 1 0 0) is the diagonalmatrix inwhich all diagonal entries are zero except those in the (119903
1+sdot sdot sdot+
119903119894minus1
+ 1)st to the (1199031+ sdot sdot sdot + 119903
119894)th rows
Similar to Lemma 9 we have the following
Lemma 10 Let 1198601 119860
119898119899isin T119898otimes T119899be idempotent
matrices of rank-1 such that for any 119894 = 119895 isin [1119898119899] 119860119894+ 119860119895is
idempotent Then there exist a permutation 120587 on [1 119898119899] andan invertible matrix 119875 isin T
119898otimesT119899such that
119860119894= 119875119864(119898119899)
120587(119894)120587(119894)119875minus1
119894 = 1 119898119899 (14)
Proof By119860119894isin T119898otimesT119899being an idempotentmatrix of rank-
1 we can assume
119860119894= 119864(119898119899)
120587(119894)120587(119894)+ 119861119894 119894 = 1 119898119899 (15)
where 119861119894isin T119898otimes T119899with zero diagonal entries It follows
from 119860119894+ 119860119895 forall119894 = 119895 being is idempotent that 120587(119894) = 120587(119895)
forall119894 = 119895 Hence 120587 is a permutation on [1 119898119899] By 1198602120587minus1
(1)=
119860120587minus1
(1) we can see 119860
120587minus1
(1)= 119864(119898119899)
11+ Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896 Let
1198751= 119868119898119899
minus Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896isin T119898otimes T119899 then 119875
minus1
1= 119868119898119899
+
Σ(119898119899)
119896=21205821119896119864(119898119899)
1119896and
119860120587minus1
(1)= 1198751119864(119898119899)
11119875minus1
1 (16)
By 119860120587minus1
(1)+ 119860
120587minus1
(2)being idempotent we obtain
119875minus1
1119860120587minus1
(2)1198751
= 119864(119898119899)
22+ Σ(119898119899)
119896=31205822119896119864(119898119899)
2119896 Let 119875
2=
119868119898119899
minus Σ(119898119899)
119896=31205822119896119864(119898119899)
2119896 then
119860120587minus1
(1)= 11987521198751119864(119898119899)
11119875minus1
1119875minus1
2
119860120587minus1
(2)= 11987521198751119864(119898119899)
22119875minus1
1119875minus1
2
(17)
Continuing to do this we can find 1198753 119875
119898119899 Let 119875 =
119875119898119899119875119898119899minus1
sdot sdot sdot 11987521198751isin T119898otimesT119899 then
119860120587minus1
(119894)= 119875119864(119898119899)
119894119894119875minus1
119894 = 1 119898119899 (18)
This completes the proof
3 The Main Result
Themain result of this paper is as follows
Theorem 11 Suppose 119898 119899 ge 2 are positive integers and 120601
is an injective linear map on T119898otimes T119899 Then 120601 (119860 otimes 119861)
4 Abstract and Applied Analysis
is an idempotent matrix in T119898otimes T119899whenever 119860 otimes 119861 is an
idempotent matrix in T119898otimes T119899if and only if there exists an
invertible matrix 119875 isin T119898otimesT119899such that
120601 (119860 otimes 119861) = 119875 (1205851(119860) otimes 120585
2(119861)) 119875
minus1
forall119860 isin T119898 119861 isin T
119899
(i)
or when119898 = 119899
120601 (119860 otimes 119861) = 119875 (1205851(119861) otimes 120585
2(119860)) 119875
minus1
forall119860 isin T119898 119861 isin T
119898
(ii)
where for 119894 = 1 2 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869
Proof The sufficiency is obvious We will prove the necessityby the following six steps
Step 1 There exist a permutation 120587 on [1 119898119899] and aninvertible matrix 119875 isin T
119898otimesT119899such that
120601 (119864(119898119899)
119896119896) = 119875119864
(119898119899)
120587(119896)120587(119896)119875minus1
forall119896 isin [1119898119899] (19)
Proof of Step 1 By Lemma 10 we only need to prove that
rank 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) = 1 forall119894 isin [1119898] 119895 isin [1 119899] (20)
And for any (119894 119895) = (119906 V) isin [1119898] times [1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119906119906otimes 119864(119899)
VV ) is an idempotentmatrix(21)
It follows from 119864(119898)
11otimes 119868119899 119864
(119898)
119898119898otimes 119868119899and (119864(119898)
119894119894+119864(119898)
119895119895) otimes
119868119899forall119894 = 119895 isin [1119898] are idempotent matrices in T
119898otimes T119899that
120601 (119864(119898)
11otimes119868119899) 120601 (119864
(119898)
119898119898otimes119868119899) and 120601 (119864(119898)
119894119894otimes119868119899)+120601 (119864
(119898)
119895119895otimes119868119899)
forall119894 = 119895 isin [1119898] are idempotent matrices We obtain by usingLemma 9 that there exists an invertible matrix 119875
1isin M119898119899
such that
120601 (119864(119898)
119894119894otimes 119868119899) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119868119903119894
0
0 0 0119898119899minus119904minus119903
119894
]
]
]
]
]
119875minus1
1 (22)
where 119903119894= rank 120601 (119864(119898)
119894119894otimes 119868119899) and 119904
119894= 1199031+ sdot sdot sdot + 119903
119894minus1 For any
119895 isin [1 119899] it follows from 119864(119898)
119894119894otimes 119864(119899)
119895119895and 119864(119898)
119894119894otimes (119868119899minus 119864(119899)
119895119895)
being idempotent matrices in T119898otimes T119899that 120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895)
and 120601 (119864(119898)119894119894
otimes 119868119899) minus 120601 (119864
(119898)
119894119894otimes 119864(119899)
119895119895) are idempotent matrices
we obtain by (22) and Lemma 6 that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119883119895
0
0 0 0119898119899minus119904
119894
minus119903119894
]
]
]
]
]
119875minus1
1
forall119895 isin [1 119899]
(23)
where 119883119895isin M119903119894
is an idempotent matrix For any 119895 = 119897 isin
[1 119899]119864(119898)119894119894
otimes(119864(119899)
119895119895+119864(119899)
119897119897) is an idempotentmatrix inT
119898otimesT119899
we have 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119894119894otimes 119864(119899)
119897119897) is an idempotent
matrix It follows from (23) that 119883119895+ 119883119897is an idempotent
matrix If 119903119894lt 119899 by Lemma 9 we can obtain that there exists
some 1198950isin [1 119899] such that 119883
1198950
= 0 This together with(23) implies 120601 (119864(119898)
119894119894otimes 119864(119899)
1198950
1198950
) = 0 which is a contradictionto the fact that 120601 is injective Hence 119903
119894= 119899 forall119894 isin [1 119899] By
Lemma 9 there exists an invertible matrix119876119894isin M119899such that
119883119895= 119876119894119864(119899)
119895119895119876minus1
119894 forall119895 isin [1 119899] Let 119876 = diag (119876
1 119876
119898) it
follows from (23) that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751119876(119864(119898)
119894119894otimes 119864(119899)
119895119895)119876minus1
119875minus1
1
forall119894 isin [1119898] 119896 isin [1 119899]
(24)
Hence (20) and (21) hold This completes the proof of Step 1By Step 1 we may assume that for any 119894 isin [1 119898] 119895 isin
[1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898119899)
120587(120588(119894119895))120587(120588(119894119895)) (25)
From this together with (3) we can also write
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864
(119898)
120590(120587(120588(119894119895)))120590(120587(120588(119894119895)))
otimes 119864(119899)
120591(120587(120588(119894119895)))120591(120587(120588(119894119895)))
(26)
Step 2(i) For any 119894 isin [1 119898] 1198951 1198952
isin [1 119899] (120590(120587(120588(1198941198951))) 120591(120587(120588(119894 119895
1)))) and (120590(120587(120588(119894 119895
2))) 120591(120587(120588(119894 119895
2)))) are
comparable(ii) For any 119894
1 1198942isin [1119898] 119895 isin [1 119899] (120590(120587(120588(119894
1 119895)))
120591(120587(120588(1198941 119895)))) and (120590(120587(120588(119894
2 119895))) 120591(120587(120588(119894
2 119895)))) are compa-
rableProof of Step 2 (i) Suppose there exist some 119894
0isin [1119898] and
1198951lt 1198952isin [1 119899] such that (120590(120587(120588(119894
0 1198951))) 120591(120587(120588(119894
0 1198951)))) and
(120590(120587(120588(1198940 1198952))) 120591(120587(120588(119894
0 1198952)))) are not comparable Without
loss of generality we may assume that
120590 (120587 (120588 (1198940 1198951))) lt 120590 (120587 (120588 (119894
0 1198952)))
120591 (120587 (120588 (1198940 1198951))) gt 120591 (120587 (120588 (119894
0 1198952)))
(27)
It follows that
120587 (120588 (1198940 1198951)) = (120590 (120587 (120588 (119894
0 1198951))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198951)))
le (120590 (120587 (120588 (1198940 1198951))) minus 1) 119899 + 119899
le (120590 (120587 (120588 (1198940 1198951)))) 119899
le (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899
lt (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198952)))
= 120587 (120588 (1198940 1198952))
(28)
Abstract and Applied Analysis 5
For any 119909 isin C by 119864(119898)1198940
1198940
otimes(119864(119899)
1198951
1198951
+119909119864(119899)
1198951
1198952
) and 119864(119898)1198940
1198940
otimes(119864(119899)
1198952
1198952
+
119909119864(119899)
1198951
1198952
) being idempotent matrices inT119898otimesT119899 we obtain by
(25) that
119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198951
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
119864(119898119899)
120587(120588(1198940
1198952
))120587(120588(1198940
1198952
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
(29)
are idempotent matrices inT119898otimesT119899 hence by Lemma 8
120601 (119864(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
) = 120582119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198952
)) (30)
From (27) by Lemma 4 we obtain that 120582 = 0 which is acontradiction to the fact that 120601 is injective Using a similarmethod we may prove (ii) holds This completes the proof ofStep 2Note It is easy to see that 120588 is a bijective map from [1 119898] times
[1 119899] to [1 119898119899] with 120588minus1
119896 997891rarr (120590(119896) 120591(119896)) from [1 119898119899]
to [1 119898] times [1 119899] This together with 120587 is a permutation on[1 119898119899] we obtain that
(120590 (120587120588 (119894 119895)) 120591 (120587120588 (119894 119895))) 119894 isin [1 119898] 119895 isin [1 119899]
= [1119898] times [1 119899]
(31)
Let 119886119894119895= (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) then [119886
119894119895] forms an 119898 times 119899
matrix containing all elements of [1 119898]times[1 119899] Step 2 impliesthat every two elements of [119886
119894119895] in the same row and column
are comparable respectivelyThus applying Lemma 5 to [119886119894119895]
we conclude that one of (I)ndash(IV) holds If (I) holds thenfor any but fixed 119894 isin [1 119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895)))119895 = 1 119899 are in the same row that is 120590(120587(120588(119894 119895))) =
120590(120587(120588(119894 1))) forall119895 isin [1 119899] and 120591(120587120588(119894 119895)) 119895 = 1 119899 =
[1 119899] hence it follows from (26) that
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
120590(120587(120588(1198941)))120590(120587(120588(1198941)))otimes 119868119899 (32)
Similarly if (III) holds then 119898 = 119899 and for any but fixed119895 isin [1119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) 119894 = 1 119898 are in thesame row that is 120590(120587120588(119894 119895)) = 120590(120587120588(1 119895)) forall119894 isin [1119898] and120591(120587120588(119894 119895)) 119894 = 1 119898 = [1119898] hence it follows from(26) that
120601 (119868119898otimes 119864(119898)
119895119895) = 119864(119898)
120590(120587(120588(1119895)))120590(120587(120588(1119895)))otimes 119868119898 (33)
We claim that (II) and (IV) do not hold Indeed if (II)holds for convenience we assume 119898 ge 3 and we firstconsider the special case 119880 = 119868
119898 119881 = 119868
119899in (II) (one can
use a similar method to prove the case of 119899 ge 3) Thus by(26) we have
120601 (119864(119898)
11otimes 119864(119898)
11) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119898119898otimes 119864(119899)
119899119899) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(34)
Since for any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
11+
119909119864(119899)
12) and (119864
(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
22+ 119909119864(119899)
12) are
idempotent matrices inT119898otimesT119899 we obtain by (34) that
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
119864(119898)
22otimes 119864(119899)
11+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
11otimes 119864(119899)
22+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
(35)
are idempotent matrices in T119898otimes T119899 This together with
Lemma 8 implies
120601 (119864(119898)
12otimes 119864(119899)
11) = 120582119864
(119898119899)
(119899+1)119898119899= 0
120601 (119864(119898)
11otimes 119864(119899)
12) = 120583119864
(119898119899)
2119898119899= 0
(36)
For any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes (119864
(119899)
11+119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 Thus by (34) and (36)
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ119896120573119896119864(119898119899)
119896119898119899 (37)
Similarly since for any 119909 isin C (119864(119898)22
+ 119909119864(119898)
12) otimes 119864(119899)
22 (119864(119898)11
+
119909119864(119898)
12) otimes 119864(119899)
22and 119864(119898)
22otimes (119864(119899)
22+ 119909119864(119899)
12) 119864(119898)22
otimes (119864(119899)
11+ 119909119864(119899)
12)
are idempotent matrices inT119898otimesT119899 we have
120601 (119864(119898)
12otimes 119864(119899)
22) = 1205821015840
119864(119898119899)
2(119899+2)= 0
120601 (119864(119898)
22otimes 119864(119899)
12) = 1205831015840
119864(119898119899)
(119899+1)(119899+2)= 0
(38)
For any 119909 isin C (119864(119898)22
+ 119864(119898)
12) otimes (119864
(119899)
22+ 119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 we obtain
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ1198961205731015840
119896119864(119898119899)
119896(119899+2) (39)
It follows from (37) and (39) that 120601 (119864(119898)12
otimes119864(119899)
12) = 0 which is
a contradiction to that 120601 is injectiveFor general case by (II) we can choose a permutation
1199011 119901
119898of [1119898] and a permutation 119902
1 119902
119899of [1 119899] such
that
(120590 (120587 (120588 (1199011 1199021))) 120591 (120587 (120588 (119901
1 1199021)))) = (119898 119899)
(120590 (120587 (120588 (119901119898 119902119899))) 120591 (120587 (120588 (119901
119898 119902119899)))) = (1 1)
(120590 (120587 (120588 (119901119894 119902119895))) 120591 (120587 (120588 (119901
119894 119902119895)))) = (119894 119895)
forall (119894 119895) = (1 1) (119898 119899)
(40)
From this together with (26) we obtain
120601 (119864(119898)
1199011
1199011
otimes 119864(119898)
1199021
1199021
) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119901119898
119901119898
otimes 119864(119899)
119902119899
119902119899
) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119901119894
119901119894
otimes 119864(119899)
119902119895
119902119895
) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(41)
6 Abstract and Applied Analysis
Using a similar method as the above we can drive a contra-diction Similarly we may prove (IV) does not hold
If (32) holds we may assume
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
119892(119894)119892(119894)otimes 119868119899 (42)
where 119892 is a permutation on [1 119898] If (33) holds we mayassume
120601 (119868119898otimes 119864(119898)
119894119894) = 119864(119898)
119892(119894)119892(119894)otimes 119868119898 (43)
We next assume (42) to prove (i) of theorem holds and onecan use similarmethods to prove (ii) of theorem if (43) holdsStep 3 There exists an invertible matrix 119875 isin T
119898otimesT119899such
that
120601 (119864(119898)
119894119894otimes 119883) = 119875 (119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883)) 119875
minus1
forall119894 isin [1119898] 119883 isin T119899
(44)
where for 119894 isin [1119898] 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869Proof of Step 3 For any idempotent matrix 119860 isin T
119899 since
119864(119898)
119894119894otimes119860 and 119864(119898)
119894119894otimes(119868119899minus119860) are idempotent matrices inT
119898otimes
T119899 we obtain by (42) that
120601 (119864(119898)
119894119894otimes 119860) 119864
(119898)
119892(119894)119892(119894)otimes 119868119899minus 120601 (119864
(119898)
119894119894otimes 119860) (45)
are idempotent matrices inT119898otimesT119899 By Lemma 7 we have
120601 (119864(119898)
119894119894otimes 119860) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119860) (46)
where 120595119894(119860) isin T
119899is an idempotent matrix By the
arbitrariness of119860 we can expand120595119894to be a linearmap onT
119899
Hence
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119883) forall119883 isin T
119899 (47)
It is easy to see that120595119894is injective and preserving idempotents
Thus by Lemma 3 there exists an invertible 119875119892(119894)
isin T119899such
that 120595119894(119883) = 119875
119892(119894)120585119894(119883)119875minus1
119892(119894) where 120585
119894(119883) = 119883 or 120585
119894(119883) =
119869119883119905
119869 Let 119875 = diag (1198751 119875
119898) isin T
119898otimesT119899 we complete the
proof of this stepBy Step 3 we may assume
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883) forall119894 isin [1119898] 119883 isin T
119899
(48)
Step 4 119892 (119894) = 119894 or 119892(119894) = 119898 minus 119894 + 1Proof of Step 4 If 119898 = 2 then this claim is clear For 119898 ge 3we prove that if 119894 lt 119895 lt 119896 then 119892 (119894) lt 119892 (119895) lt 119892 (119896) or 119892 (119894) gt119892 (119895) gt 119892 (119896) Otherwise we assume 119892 (119894) lt 119892 (119896) lt 119892 (119895)
(other cases can be proven by using similar methods) Sincefor any 119909 isin C (119864(119898)
119894119894+ 119909119864(119898)
119894119895) otimes 119868119899and (119864(119898)
119895119895+ 119909119864(119898)
119894119895) otimes 119868119899are
idempotent matrices inT119898otimesT119899 we have by using (48) that
119864(119898)
119892(119894)119892(119894)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899) and 119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899)
are idempotent matrices in 119879119898otimes 119879119899 This together with
Lemma 8 implies
120601 (119864(119898)
119894119895otimes 119868119899) = 119864(119898)
119892(119894)119892(119895)otimes 119860 for some119860 = 0 isin T
119899 (49)
Similarly
120601 (119864(119898)
119894119896otimes 119868119899) = 119864(119898)
119892(119894)119892(119896)otimes 119861 for some119861 = 0 isin T
119899
120601 (119864(119898)
119895119896otimes 119868119899) = 119864(119898)
119892(119896)119892(119895)otimes 119862 for some119862 = 0 isin T
119899
(50)
By (119864(119898)119895119895
+119864(119898)
119894119895+119864(119898)
119894119896+119864(119898)
119895119896)otimes119868119899being an idempotent matrix
inT119898otimesT119899 we obtain by using (48) (49) and (50) that
119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119864(119898)
119892(119894)119892(119895)otimes 119860 + 119864
(119898)
119892(119894)119892(119896)otimes 119861 + 119864
(119898)
119892(119896)119892(119895)otimes 119862
(51)
is an idempotent matrix inT119898otimesT119899 that is
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
2
=
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
(52)
This implies 119860 = 0 which is a contradiction Hence wecomplete the proof of Step 4Step 5 For any 119894 lt 119895 120585
119894= 120585119895 and there exists 120582
119894119895= 0 such that
120601 (119864(119898)
119894119895otimes 119883) = 120582
119894119895119864(119898)
119892(119894)119892(119895)otimes 120585119894(119883) forall119894 = 119895 119883 isin T
119899
(53)
Proof of Step 5 We prove the case of 119892 (119894) = 119894 (one can use asimilar method to prove the case of 119892 (119894) = 119898 minus 119894 + 1) Hence
120601 (119864(119898)
119894119894otimes 119864(119899)
119896119896) = 119864(119898)
119894119894otimes 119864(119899)
119896119896 forall119894 isin [1119898] 119896 isin [1 119899]
(54)
Without loss of generality we may assume 119894 = 1 119895 = 2 Sincefor any 119909 isin C (119864(119898)
11+ 119909119864(119898)
12) otimes 119864(119899)
119896119896and (119864(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
119896119896
are idempotent matrices inT119898otimesT119899 we obtain by (54) that
119864(119898)
11otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) and119864(119898)
22otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) are
idempotentmatrices in119879119898otimes119879119899This together with Lemma 8
implies 120601(119864(119898)12
otimes 119864(119899)
119896119896) = 120582119896119864(119898)
12otimes 119864(119899)
119896119896 where 120582
119896= 0 Let Λ =
diag (1205821 120582
119898) then
120601 (119864(119898)
12otimes 119868119899) =
[
[
0 Λ
0 0
]
]
oplus 0 (55)
Let119876 = [119868119898
minusΛ
0 119868119898
]oplus119868(119898minus2)119899
isin T119898otimesT119899 by (55) one can obtain
120601 ((119864(119898)
11+ 119864(119898)
12) otimes 119868119899) =
[
[
119868119899Λ
0 0
]
]
oplus 0
= 119876 (119864(119898)
11otimes 119868119899)119876minus1
120601 ((119864(119898)
22minus 119864(119898)
12) otimes 119868119899) =
[
[
0 minusΛ
0 119868119899
]
]
oplus 0
= 119876 (119864(119898)
22otimes 119868119899)119876minus1
(56)
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
is an idempotent matrix in T119898otimes T119899whenever 119860 otimes 119861 is an
idempotent matrix in T119898otimes T119899if and only if there exists an
invertible matrix 119875 isin T119898otimesT119899such that
120601 (119860 otimes 119861) = 119875 (1205851(119860) otimes 120585
2(119861)) 119875
minus1
forall119860 isin T119898 119861 isin T
119899
(i)
or when119898 = 119899
120601 (119860 otimes 119861) = 119875 (1205851(119861) otimes 120585
2(119860)) 119875
minus1
forall119860 isin T119898 119861 isin T
119898
(ii)
where for 119894 = 1 2 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869
Proof The sufficiency is obvious We will prove the necessityby the following six steps
Step 1 There exist a permutation 120587 on [1 119898119899] and aninvertible matrix 119875 isin T
119898otimesT119899such that
120601 (119864(119898119899)
119896119896) = 119875119864
(119898119899)
120587(119896)120587(119896)119875minus1
forall119896 isin [1119898119899] (19)
Proof of Step 1 By Lemma 10 we only need to prove that
rank 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) = 1 forall119894 isin [1119898] 119895 isin [1 119899] (20)
And for any (119894 119895) = (119906 V) isin [1119898] times [1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119906119906otimes 119864(119899)
VV ) is an idempotentmatrix(21)
It follows from 119864(119898)
11otimes 119868119899 119864
(119898)
119898119898otimes 119868119899and (119864(119898)
119894119894+119864(119898)
119895119895) otimes
119868119899forall119894 = 119895 isin [1119898] are idempotent matrices in T
119898otimes T119899that
120601 (119864(119898)
11otimes119868119899) 120601 (119864
(119898)
119898119898otimes119868119899) and 120601 (119864(119898)
119894119894otimes119868119899)+120601 (119864
(119898)
119895119895otimes119868119899)
forall119894 = 119895 isin [1119898] are idempotent matrices We obtain by usingLemma 9 that there exists an invertible matrix 119875
1isin M119898119899
such that
120601 (119864(119898)
119894119894otimes 119868119899) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119868119903119894
0
0 0 0119898119899minus119904minus119903
119894
]
]
]
]
]
119875minus1
1 (22)
where 119903119894= rank 120601 (119864(119898)
119894119894otimes 119868119899) and 119904
119894= 1199031+ sdot sdot sdot + 119903
119894minus1 For any
119895 isin [1 119899] it follows from 119864(119898)
119894119894otimes 119864(119899)
119895119895and 119864(119898)
119894119894otimes (119868119899minus 119864(119899)
119895119895)
being idempotent matrices in T119898otimes T119899that 120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895)
and 120601 (119864(119898)119894119894
otimes 119868119899) minus 120601 (119864
(119898)
119894119894otimes 119864(119899)
119895119895) are idempotent matrices
we obtain by (22) and Lemma 6 that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751
[
[
[
[
[
0119904119894
0 0
0 119883119895
0
0 0 0119898119899minus119904
119894
minus119903119894
]
]
]
]
]
119875minus1
1
forall119895 isin [1 119899]
(23)
where 119883119895isin M119903119894
is an idempotent matrix For any 119895 = 119897 isin
[1 119899]119864(119898)119894119894
otimes(119864(119899)
119895119895+119864(119899)
119897119897) is an idempotentmatrix inT
119898otimesT119899
we have 120601 (119864(119898)119894119894
otimes 119864(119899)
119895119895) + 120601 (119864
(119898)
119894119894otimes 119864(119899)
119897119897) is an idempotent
matrix It follows from (23) that 119883119895+ 119883119897is an idempotent
matrix If 119903119894lt 119899 by Lemma 9 we can obtain that there exists
some 1198950isin [1 119899] such that 119883
1198950
= 0 This together with(23) implies 120601 (119864(119898)
119894119894otimes 119864(119899)
1198950
1198950
) = 0 which is a contradictionto the fact that 120601 is injective Hence 119903
119894= 119899 forall119894 isin [1 119899] By
Lemma 9 there exists an invertible matrix119876119894isin M119899such that
119883119895= 119876119894119864(119899)
119895119895119876minus1
119894 forall119895 isin [1 119899] Let 119876 = diag (119876
1 119876
119898) it
follows from (23) that
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 1198751119876(119864(119898)
119894119894otimes 119864(119899)
119895119895)119876minus1
119875minus1
1
forall119894 isin [1119898] 119896 isin [1 119899]
(24)
Hence (20) and (21) hold This completes the proof of Step 1By Step 1 we may assume that for any 119894 isin [1 119898] 119895 isin
[1 119899]
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898119899)
120587(120588(119894119895))120587(120588(119894119895)) (25)
From this together with (3) we can also write
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864
(119898)
120590(120587(120588(119894119895)))120590(120587(120588(119894119895)))
otimes 119864(119899)
120591(120587(120588(119894119895)))120591(120587(120588(119894119895)))
(26)
Step 2(i) For any 119894 isin [1 119898] 1198951 1198952
isin [1 119899] (120590(120587(120588(1198941198951))) 120591(120587(120588(119894 119895
1)))) and (120590(120587(120588(119894 119895
2))) 120591(120587(120588(119894 119895
2)))) are
comparable(ii) For any 119894
1 1198942isin [1119898] 119895 isin [1 119899] (120590(120587(120588(119894
1 119895)))
120591(120587(120588(1198941 119895)))) and (120590(120587(120588(119894
2 119895))) 120591(120587(120588(119894
2 119895)))) are compa-
rableProof of Step 2 (i) Suppose there exist some 119894
0isin [1119898] and
1198951lt 1198952isin [1 119899] such that (120590(120587(120588(119894
0 1198951))) 120591(120587(120588(119894
0 1198951)))) and
(120590(120587(120588(1198940 1198952))) 120591(120587(120588(119894
0 1198952)))) are not comparable Without
loss of generality we may assume that
120590 (120587 (120588 (1198940 1198951))) lt 120590 (120587 (120588 (119894
0 1198952)))
120591 (120587 (120588 (1198940 1198951))) gt 120591 (120587 (120588 (119894
0 1198952)))
(27)
It follows that
120587 (120588 (1198940 1198951)) = (120590 (120587 (120588 (119894
0 1198951))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198951)))
le (120590 (120587 (120588 (1198940 1198951))) minus 1) 119899 + 119899
le (120590 (120587 (120588 (1198940 1198951)))) 119899
le (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899
lt (120590 (120587 (120588 (1198940 1198952))) minus 1) 119899 + 120591 (120587 (120588 (119894
0 1198952)))
= 120587 (120588 (1198940 1198952))
(28)
Abstract and Applied Analysis 5
For any 119909 isin C by 119864(119898)1198940
1198940
otimes(119864(119899)
1198951
1198951
+119909119864(119899)
1198951
1198952
) and 119864(119898)1198940
1198940
otimes(119864(119899)
1198952
1198952
+
119909119864(119899)
1198951
1198952
) being idempotent matrices inT119898otimesT119899 we obtain by
(25) that
119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198951
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
119864(119898119899)
120587(120588(1198940
1198952
))120587(120588(1198940
1198952
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
(29)
are idempotent matrices inT119898otimesT119899 hence by Lemma 8
120601 (119864(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
) = 120582119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198952
)) (30)
From (27) by Lemma 4 we obtain that 120582 = 0 which is acontradiction to the fact that 120601 is injective Using a similarmethod we may prove (ii) holds This completes the proof ofStep 2Note It is easy to see that 120588 is a bijective map from [1 119898] times
[1 119899] to [1 119898119899] with 120588minus1
119896 997891rarr (120590(119896) 120591(119896)) from [1 119898119899]
to [1 119898] times [1 119899] This together with 120587 is a permutation on[1 119898119899] we obtain that
(120590 (120587120588 (119894 119895)) 120591 (120587120588 (119894 119895))) 119894 isin [1 119898] 119895 isin [1 119899]
= [1119898] times [1 119899]
(31)
Let 119886119894119895= (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) then [119886
119894119895] forms an 119898 times 119899
matrix containing all elements of [1 119898]times[1 119899] Step 2 impliesthat every two elements of [119886
119894119895] in the same row and column
are comparable respectivelyThus applying Lemma 5 to [119886119894119895]
we conclude that one of (I)ndash(IV) holds If (I) holds thenfor any but fixed 119894 isin [1 119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895)))119895 = 1 119899 are in the same row that is 120590(120587(120588(119894 119895))) =
120590(120587(120588(119894 1))) forall119895 isin [1 119899] and 120591(120587120588(119894 119895)) 119895 = 1 119899 =
[1 119899] hence it follows from (26) that
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
120590(120587(120588(1198941)))120590(120587(120588(1198941)))otimes 119868119899 (32)
Similarly if (III) holds then 119898 = 119899 and for any but fixed119895 isin [1119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) 119894 = 1 119898 are in thesame row that is 120590(120587120588(119894 119895)) = 120590(120587120588(1 119895)) forall119894 isin [1119898] and120591(120587120588(119894 119895)) 119894 = 1 119898 = [1119898] hence it follows from(26) that
120601 (119868119898otimes 119864(119898)
119895119895) = 119864(119898)
120590(120587(120588(1119895)))120590(120587(120588(1119895)))otimes 119868119898 (33)
We claim that (II) and (IV) do not hold Indeed if (II)holds for convenience we assume 119898 ge 3 and we firstconsider the special case 119880 = 119868
119898 119881 = 119868
119899in (II) (one can
use a similar method to prove the case of 119899 ge 3) Thus by(26) we have
120601 (119864(119898)
11otimes 119864(119898)
11) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119898119898otimes 119864(119899)
119899119899) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(34)
Since for any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
11+
119909119864(119899)
12) and (119864
(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
22+ 119909119864(119899)
12) are
idempotent matrices inT119898otimesT119899 we obtain by (34) that
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
119864(119898)
22otimes 119864(119899)
11+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
11otimes 119864(119899)
22+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
(35)
are idempotent matrices in T119898otimes T119899 This together with
Lemma 8 implies
120601 (119864(119898)
12otimes 119864(119899)
11) = 120582119864
(119898119899)
(119899+1)119898119899= 0
120601 (119864(119898)
11otimes 119864(119899)
12) = 120583119864
(119898119899)
2119898119899= 0
(36)
For any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes (119864
(119899)
11+119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 Thus by (34) and (36)
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ119896120573119896119864(119898119899)
119896119898119899 (37)
Similarly since for any 119909 isin C (119864(119898)22
+ 119909119864(119898)
12) otimes 119864(119899)
22 (119864(119898)11
+
119909119864(119898)
12) otimes 119864(119899)
22and 119864(119898)
22otimes (119864(119899)
22+ 119909119864(119899)
12) 119864(119898)22
otimes (119864(119899)
11+ 119909119864(119899)
12)
are idempotent matrices inT119898otimesT119899 we have
120601 (119864(119898)
12otimes 119864(119899)
22) = 1205821015840
119864(119898119899)
2(119899+2)= 0
120601 (119864(119898)
22otimes 119864(119899)
12) = 1205831015840
119864(119898119899)
(119899+1)(119899+2)= 0
(38)
For any 119909 isin C (119864(119898)22
+ 119864(119898)
12) otimes (119864
(119899)
22+ 119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 we obtain
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ1198961205731015840
119896119864(119898119899)
119896(119899+2) (39)
It follows from (37) and (39) that 120601 (119864(119898)12
otimes119864(119899)
12) = 0 which is
a contradiction to that 120601 is injectiveFor general case by (II) we can choose a permutation
1199011 119901
119898of [1119898] and a permutation 119902
1 119902
119899of [1 119899] such
that
(120590 (120587 (120588 (1199011 1199021))) 120591 (120587 (120588 (119901
1 1199021)))) = (119898 119899)
(120590 (120587 (120588 (119901119898 119902119899))) 120591 (120587 (120588 (119901
119898 119902119899)))) = (1 1)
(120590 (120587 (120588 (119901119894 119902119895))) 120591 (120587 (120588 (119901
119894 119902119895)))) = (119894 119895)
forall (119894 119895) = (1 1) (119898 119899)
(40)
From this together with (26) we obtain
120601 (119864(119898)
1199011
1199011
otimes 119864(119898)
1199021
1199021
) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119901119898
119901119898
otimes 119864(119899)
119902119899
119902119899
) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119901119894
119901119894
otimes 119864(119899)
119902119895
119902119895
) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(41)
6 Abstract and Applied Analysis
Using a similar method as the above we can drive a contra-diction Similarly we may prove (IV) does not hold
If (32) holds we may assume
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
119892(119894)119892(119894)otimes 119868119899 (42)
where 119892 is a permutation on [1 119898] If (33) holds we mayassume
120601 (119868119898otimes 119864(119898)
119894119894) = 119864(119898)
119892(119894)119892(119894)otimes 119868119898 (43)
We next assume (42) to prove (i) of theorem holds and onecan use similarmethods to prove (ii) of theorem if (43) holdsStep 3 There exists an invertible matrix 119875 isin T
119898otimesT119899such
that
120601 (119864(119898)
119894119894otimes 119883) = 119875 (119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883)) 119875
minus1
forall119894 isin [1119898] 119883 isin T119899
(44)
where for 119894 isin [1119898] 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869Proof of Step 3 For any idempotent matrix 119860 isin T
119899 since
119864(119898)
119894119894otimes119860 and 119864(119898)
119894119894otimes(119868119899minus119860) are idempotent matrices inT
119898otimes
T119899 we obtain by (42) that
120601 (119864(119898)
119894119894otimes 119860) 119864
(119898)
119892(119894)119892(119894)otimes 119868119899minus 120601 (119864
(119898)
119894119894otimes 119860) (45)
are idempotent matrices inT119898otimesT119899 By Lemma 7 we have
120601 (119864(119898)
119894119894otimes 119860) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119860) (46)
where 120595119894(119860) isin T
119899is an idempotent matrix By the
arbitrariness of119860 we can expand120595119894to be a linearmap onT
119899
Hence
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119883) forall119883 isin T
119899 (47)
It is easy to see that120595119894is injective and preserving idempotents
Thus by Lemma 3 there exists an invertible 119875119892(119894)
isin T119899such
that 120595119894(119883) = 119875
119892(119894)120585119894(119883)119875minus1
119892(119894) where 120585
119894(119883) = 119883 or 120585
119894(119883) =
119869119883119905
119869 Let 119875 = diag (1198751 119875
119898) isin T
119898otimesT119899 we complete the
proof of this stepBy Step 3 we may assume
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883) forall119894 isin [1119898] 119883 isin T
119899
(48)
Step 4 119892 (119894) = 119894 or 119892(119894) = 119898 minus 119894 + 1Proof of Step 4 If 119898 = 2 then this claim is clear For 119898 ge 3we prove that if 119894 lt 119895 lt 119896 then 119892 (119894) lt 119892 (119895) lt 119892 (119896) or 119892 (119894) gt119892 (119895) gt 119892 (119896) Otherwise we assume 119892 (119894) lt 119892 (119896) lt 119892 (119895)
(other cases can be proven by using similar methods) Sincefor any 119909 isin C (119864(119898)
119894119894+ 119909119864(119898)
119894119895) otimes 119868119899and (119864(119898)
119895119895+ 119909119864(119898)
119894119895) otimes 119868119899are
idempotent matrices inT119898otimesT119899 we have by using (48) that
119864(119898)
119892(119894)119892(119894)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899) and 119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899)
are idempotent matrices in 119879119898otimes 119879119899 This together with
Lemma 8 implies
120601 (119864(119898)
119894119895otimes 119868119899) = 119864(119898)
119892(119894)119892(119895)otimes 119860 for some119860 = 0 isin T
119899 (49)
Similarly
120601 (119864(119898)
119894119896otimes 119868119899) = 119864(119898)
119892(119894)119892(119896)otimes 119861 for some119861 = 0 isin T
119899
120601 (119864(119898)
119895119896otimes 119868119899) = 119864(119898)
119892(119896)119892(119895)otimes 119862 for some119862 = 0 isin T
119899
(50)
By (119864(119898)119895119895
+119864(119898)
119894119895+119864(119898)
119894119896+119864(119898)
119895119896)otimes119868119899being an idempotent matrix
inT119898otimesT119899 we obtain by using (48) (49) and (50) that
119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119864(119898)
119892(119894)119892(119895)otimes 119860 + 119864
(119898)
119892(119894)119892(119896)otimes 119861 + 119864
(119898)
119892(119896)119892(119895)otimes 119862
(51)
is an idempotent matrix inT119898otimesT119899 that is
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
2
=
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
(52)
This implies 119860 = 0 which is a contradiction Hence wecomplete the proof of Step 4Step 5 For any 119894 lt 119895 120585
119894= 120585119895 and there exists 120582
119894119895= 0 such that
120601 (119864(119898)
119894119895otimes 119883) = 120582
119894119895119864(119898)
119892(119894)119892(119895)otimes 120585119894(119883) forall119894 = 119895 119883 isin T
119899
(53)
Proof of Step 5 We prove the case of 119892 (119894) = 119894 (one can use asimilar method to prove the case of 119892 (119894) = 119898 minus 119894 + 1) Hence
120601 (119864(119898)
119894119894otimes 119864(119899)
119896119896) = 119864(119898)
119894119894otimes 119864(119899)
119896119896 forall119894 isin [1119898] 119896 isin [1 119899]
(54)
Without loss of generality we may assume 119894 = 1 119895 = 2 Sincefor any 119909 isin C (119864(119898)
11+ 119909119864(119898)
12) otimes 119864(119899)
119896119896and (119864(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
119896119896
are idempotent matrices inT119898otimesT119899 we obtain by (54) that
119864(119898)
11otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) and119864(119898)
22otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) are
idempotentmatrices in119879119898otimes119879119899This together with Lemma 8
implies 120601(119864(119898)12
otimes 119864(119899)
119896119896) = 120582119896119864(119898)
12otimes 119864(119899)
119896119896 where 120582
119896= 0 Let Λ =
diag (1205821 120582
119898) then
120601 (119864(119898)
12otimes 119868119899) =
[
[
0 Λ
0 0
]
]
oplus 0 (55)
Let119876 = [119868119898
minusΛ
0 119868119898
]oplus119868(119898minus2)119899
isin T119898otimesT119899 by (55) one can obtain
120601 ((119864(119898)
11+ 119864(119898)
12) otimes 119868119899) =
[
[
119868119899Λ
0 0
]
]
oplus 0
= 119876 (119864(119898)
11otimes 119868119899)119876minus1
120601 ((119864(119898)
22minus 119864(119898)
12) otimes 119868119899) =
[
[
0 minusΛ
0 119868119899
]
]
oplus 0
= 119876 (119864(119898)
22otimes 119868119899)119876minus1
(56)
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
For any 119909 isin C by 119864(119898)1198940
1198940
otimes(119864(119899)
1198951
1198951
+119909119864(119899)
1198951
1198952
) and 119864(119898)1198940
1198940
otimes(119864(119899)
1198952
1198952
+
119909119864(119899)
1198951
1198952
) being idempotent matrices inT119898otimesT119899 we obtain by
(25) that
119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198951
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
119864(119898119899)
120587(120588(1198940
1198952
))120587(120588(1198940
1198952
))+ 119909120601 (119864
(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
)
(29)
are idempotent matrices inT119898otimesT119899 hence by Lemma 8
120601 (119864(119898)
1198940
1198940
otimes 119864(119899)
1198951
1198952
) = 120582119864(119898119899)
120587(120588(1198940
1198951
))120587(120588(1198940
1198952
)) (30)
From (27) by Lemma 4 we obtain that 120582 = 0 which is acontradiction to the fact that 120601 is injective Using a similarmethod we may prove (ii) holds This completes the proof ofStep 2Note It is easy to see that 120588 is a bijective map from [1 119898] times
[1 119899] to [1 119898119899] with 120588minus1
119896 997891rarr (120590(119896) 120591(119896)) from [1 119898119899]
to [1 119898] times [1 119899] This together with 120587 is a permutation on[1 119898119899] we obtain that
(120590 (120587120588 (119894 119895)) 120591 (120587120588 (119894 119895))) 119894 isin [1 119898] 119895 isin [1 119899]
= [1119898] times [1 119899]
(31)
Let 119886119894119895= (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) then [119886
119894119895] forms an 119898 times 119899
matrix containing all elements of [1 119898]times[1 119899] Step 2 impliesthat every two elements of [119886
119894119895] in the same row and column
are comparable respectivelyThus applying Lemma 5 to [119886119894119895]
we conclude that one of (I)ndash(IV) holds If (I) holds thenfor any but fixed 119894 isin [1 119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895)))119895 = 1 119899 are in the same row that is 120590(120587(120588(119894 119895))) =
120590(120587(120588(119894 1))) forall119895 isin [1 119899] and 120591(120587120588(119894 119895)) 119895 = 1 119899 =
[1 119899] hence it follows from (26) that
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
120590(120587(120588(1198941)))120590(120587(120588(1198941)))otimes 119868119899 (32)
Similarly if (III) holds then 119898 = 119899 and for any but fixed119895 isin [1119898] all (120590(120587120588(119894 119895)) 120591(120587120588(119894 119895))) 119894 = 1 119898 are in thesame row that is 120590(120587120588(119894 119895)) = 120590(120587120588(1 119895)) forall119894 isin [1119898] and120591(120587120588(119894 119895)) 119894 = 1 119898 = [1119898] hence it follows from(26) that
120601 (119868119898otimes 119864(119898)
119895119895) = 119864(119898)
120590(120587(120588(1119895)))120590(120587(120588(1119895)))otimes 119868119898 (33)
We claim that (II) and (IV) do not hold Indeed if (II)holds for convenience we assume 119898 ge 3 and we firstconsider the special case 119880 = 119868
119898 119881 = 119868
119899in (II) (one can
use a similar method to prove the case of 119899 ge 3) Thus by(26) we have
120601 (119864(119898)
11otimes 119864(119898)
11) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119898119898otimes 119864(119899)
119899119899) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119894119894otimes 119864(119899)
119895119895) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(34)
Since for any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
11+
119909119864(119899)
12) and (119864
(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
11 119864(119898)11
otimes (119864(119899)
22+ 119909119864(119899)
12) are
idempotent matrices inT119898otimesT119899 we obtain by (34) that
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
119898119898otimes 119864(119899)
119899119899+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
119864(119898)
22otimes 119864(119899)
11+ 119909120601 (119864
(119898)
12otimes 119864(119899)
11)
119864(119898)
11otimes 119864(119899)
22+ 119909120601 (119864
(119898)
11otimes 119864(119899)
12)
(35)
are idempotent matrices in T119898otimes T119899 This together with
Lemma 8 implies
120601 (119864(119898)
12otimes 119864(119899)
11) = 120582119864
(119898119899)
(119899+1)119898119899= 0
120601 (119864(119898)
11otimes 119864(119899)
12) = 120583119864
(119898119899)
2119898119899= 0
(36)
For any 119909 isin C (119864(119898)11
+ 119909119864(119898)
12) otimes (119864
(119899)
11+119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 Thus by (34) and (36)
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ119896120573119896119864(119898119899)
119896119898119899 (37)
Similarly since for any 119909 isin C (119864(119898)22
+ 119909119864(119898)
12) otimes 119864(119899)
22 (119864(119898)11
+
119909119864(119898)
12) otimes 119864(119899)
22and 119864(119898)
22otimes (119864(119899)
22+ 119909119864(119899)
12) 119864(119898)22
otimes (119864(119899)
11+ 119909119864(119899)
12)
are idempotent matrices inT119898otimesT119899 we have
120601 (119864(119898)
12otimes 119864(119899)
22) = 1205821015840
119864(119898119899)
2(119899+2)= 0
120601 (119864(119898)
22otimes 119864(119899)
12) = 1205831015840
119864(119898119899)
(119899+1)(119899+2)= 0
(38)
For any 119909 isin C (119864(119898)22
+ 119864(119898)
12) otimes (119864
(119899)
22+ 119864(119899)
12) is an idempotent
matrix inT119898otimesT119899 we obtain
120601 (119864(119898)
12otimes 119864(119899)
12) = Σ1198961205731015840
119896119864(119898119899)
119896(119899+2) (39)
It follows from (37) and (39) that 120601 (119864(119898)12
otimes119864(119899)
12) = 0 which is
a contradiction to that 120601 is injectiveFor general case by (II) we can choose a permutation
1199011 119901
119898of [1119898] and a permutation 119902
1 119902
119899of [1 119899] such
that
(120590 (120587 (120588 (1199011 1199021))) 120591 (120587 (120588 (119901
1 1199021)))) = (119898 119899)
(120590 (120587 (120588 (119901119898 119902119899))) 120591 (120587 (120588 (119901
119898 119902119899)))) = (1 1)
(120590 (120587 (120588 (119901119894 119902119895))) 120591 (120587 (120588 (119901
119894 119902119895)))) = (119894 119895)
forall (119894 119895) = (1 1) (119898 119899)
(40)
From this together with (26) we obtain
120601 (119864(119898)
1199011
1199011
otimes 119864(119898)
1199021
1199021
) = 119864(119898)
119898119898otimes 119864(119899)
119899119899
120601 (119864(119898)
119901119898
119901119898
otimes 119864(119899)
119902119899
119902119899
) = 119864(119898)
11otimes 119864(119898)
11
120601 (119864(119898)
119901119894
119901119894
otimes 119864(119899)
119902119895
119902119895
) = 119864(119898)
119894119894otimes 119864(119899)
119895119895 forall (119894 119895) = (1 1) (119898 119899)
(41)
6 Abstract and Applied Analysis
Using a similar method as the above we can drive a contra-diction Similarly we may prove (IV) does not hold
If (32) holds we may assume
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
119892(119894)119892(119894)otimes 119868119899 (42)
where 119892 is a permutation on [1 119898] If (33) holds we mayassume
120601 (119868119898otimes 119864(119898)
119894119894) = 119864(119898)
119892(119894)119892(119894)otimes 119868119898 (43)
We next assume (42) to prove (i) of theorem holds and onecan use similarmethods to prove (ii) of theorem if (43) holdsStep 3 There exists an invertible matrix 119875 isin T
119898otimesT119899such
that
120601 (119864(119898)
119894119894otimes 119883) = 119875 (119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883)) 119875
minus1
forall119894 isin [1119898] 119883 isin T119899
(44)
where for 119894 isin [1119898] 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869Proof of Step 3 For any idempotent matrix 119860 isin T
119899 since
119864(119898)
119894119894otimes119860 and 119864(119898)
119894119894otimes(119868119899minus119860) are idempotent matrices inT
119898otimes
T119899 we obtain by (42) that
120601 (119864(119898)
119894119894otimes 119860) 119864
(119898)
119892(119894)119892(119894)otimes 119868119899minus 120601 (119864
(119898)
119894119894otimes 119860) (45)
are idempotent matrices inT119898otimesT119899 By Lemma 7 we have
120601 (119864(119898)
119894119894otimes 119860) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119860) (46)
where 120595119894(119860) isin T
119899is an idempotent matrix By the
arbitrariness of119860 we can expand120595119894to be a linearmap onT
119899
Hence
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119883) forall119883 isin T
119899 (47)
It is easy to see that120595119894is injective and preserving idempotents
Thus by Lemma 3 there exists an invertible 119875119892(119894)
isin T119899such
that 120595119894(119883) = 119875
119892(119894)120585119894(119883)119875minus1
119892(119894) where 120585
119894(119883) = 119883 or 120585
119894(119883) =
119869119883119905
119869 Let 119875 = diag (1198751 119875
119898) isin T
119898otimesT119899 we complete the
proof of this stepBy Step 3 we may assume
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883) forall119894 isin [1119898] 119883 isin T
119899
(48)
Step 4 119892 (119894) = 119894 or 119892(119894) = 119898 minus 119894 + 1Proof of Step 4 If 119898 = 2 then this claim is clear For 119898 ge 3we prove that if 119894 lt 119895 lt 119896 then 119892 (119894) lt 119892 (119895) lt 119892 (119896) or 119892 (119894) gt119892 (119895) gt 119892 (119896) Otherwise we assume 119892 (119894) lt 119892 (119896) lt 119892 (119895)
(other cases can be proven by using similar methods) Sincefor any 119909 isin C (119864(119898)
119894119894+ 119909119864(119898)
119894119895) otimes 119868119899and (119864(119898)
119895119895+ 119909119864(119898)
119894119895) otimes 119868119899are
idempotent matrices inT119898otimesT119899 we have by using (48) that
119864(119898)
119892(119894)119892(119894)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899) and 119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899)
are idempotent matrices in 119879119898otimes 119879119899 This together with
Lemma 8 implies
120601 (119864(119898)
119894119895otimes 119868119899) = 119864(119898)
119892(119894)119892(119895)otimes 119860 for some119860 = 0 isin T
119899 (49)
Similarly
120601 (119864(119898)
119894119896otimes 119868119899) = 119864(119898)
119892(119894)119892(119896)otimes 119861 for some119861 = 0 isin T
119899
120601 (119864(119898)
119895119896otimes 119868119899) = 119864(119898)
119892(119896)119892(119895)otimes 119862 for some119862 = 0 isin T
119899
(50)
By (119864(119898)119895119895
+119864(119898)
119894119895+119864(119898)
119894119896+119864(119898)
119895119896)otimes119868119899being an idempotent matrix
inT119898otimesT119899 we obtain by using (48) (49) and (50) that
119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119864(119898)
119892(119894)119892(119895)otimes 119860 + 119864
(119898)
119892(119894)119892(119896)otimes 119861 + 119864
(119898)
119892(119896)119892(119895)otimes 119862
(51)
is an idempotent matrix inT119898otimesT119899 that is
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
2
=
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
(52)
This implies 119860 = 0 which is a contradiction Hence wecomplete the proof of Step 4Step 5 For any 119894 lt 119895 120585
119894= 120585119895 and there exists 120582
119894119895= 0 such that
120601 (119864(119898)
119894119895otimes 119883) = 120582
119894119895119864(119898)
119892(119894)119892(119895)otimes 120585119894(119883) forall119894 = 119895 119883 isin T
119899
(53)
Proof of Step 5 We prove the case of 119892 (119894) = 119894 (one can use asimilar method to prove the case of 119892 (119894) = 119898 minus 119894 + 1) Hence
120601 (119864(119898)
119894119894otimes 119864(119899)
119896119896) = 119864(119898)
119894119894otimes 119864(119899)
119896119896 forall119894 isin [1119898] 119896 isin [1 119899]
(54)
Without loss of generality we may assume 119894 = 1 119895 = 2 Sincefor any 119909 isin C (119864(119898)
11+ 119909119864(119898)
12) otimes 119864(119899)
119896119896and (119864(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
119896119896
are idempotent matrices inT119898otimesT119899 we obtain by (54) that
119864(119898)
11otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) and119864(119898)
22otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) are
idempotentmatrices in119879119898otimes119879119899This together with Lemma 8
implies 120601(119864(119898)12
otimes 119864(119899)
119896119896) = 120582119896119864(119898)
12otimes 119864(119899)
119896119896 where 120582
119896= 0 Let Λ =
diag (1205821 120582
119898) then
120601 (119864(119898)
12otimes 119868119899) =
[
[
0 Λ
0 0
]
]
oplus 0 (55)
Let119876 = [119868119898
minusΛ
0 119868119898
]oplus119868(119898minus2)119899
isin T119898otimesT119899 by (55) one can obtain
120601 ((119864(119898)
11+ 119864(119898)
12) otimes 119868119899) =
[
[
119868119899Λ
0 0
]
]
oplus 0
= 119876 (119864(119898)
11otimes 119868119899)119876minus1
120601 ((119864(119898)
22minus 119864(119898)
12) otimes 119868119899) =
[
[
0 minusΛ
0 119868119899
]
]
oplus 0
= 119876 (119864(119898)
22otimes 119868119899)119876minus1
(56)
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Using a similar method as the above we can drive a contra-diction Similarly we may prove (IV) does not hold
If (32) holds we may assume
120601 (119864(119898)
119894119894otimes 119868119899) = 119864(119898)
119892(119894)119892(119894)otimes 119868119899 (42)
where 119892 is a permutation on [1 119898] If (33) holds we mayassume
120601 (119868119898otimes 119864(119898)
119894119894) = 119864(119898)
119892(119894)119892(119894)otimes 119868119898 (43)
We next assume (42) to prove (i) of theorem holds and onecan use similarmethods to prove (ii) of theorem if (43) holdsStep 3 There exists an invertible matrix 119875 isin T
119898otimesT119899such
that
120601 (119864(119898)
119894119894otimes 119883) = 119875 (119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883)) 119875
minus1
forall119894 isin [1119898] 119883 isin T119899
(44)
where for 119894 isin [1119898] 120585119894(119883) = 119883 or 120585
119894(119883) = 119869119883
119905
119869Proof of Step 3 For any idempotent matrix 119860 isin T
119899 since
119864(119898)
119894119894otimes119860 and 119864(119898)
119894119894otimes(119868119899minus119860) are idempotent matrices inT
119898otimes
T119899 we obtain by (42) that
120601 (119864(119898)
119894119894otimes 119860) 119864
(119898)
119892(119894)119892(119894)otimes 119868119899minus 120601 (119864
(119898)
119894119894otimes 119860) (45)
are idempotent matrices inT119898otimesT119899 By Lemma 7 we have
120601 (119864(119898)
119894119894otimes 119860) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119860) (46)
where 120595119894(119860) isin T
119899is an idempotent matrix By the
arbitrariness of119860 we can expand120595119894to be a linearmap onT
119899
Hence
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120595119894(119883) forall119883 isin T
119899 (47)
It is easy to see that120595119894is injective and preserving idempotents
Thus by Lemma 3 there exists an invertible 119875119892(119894)
isin T119899such
that 120595119894(119883) = 119875
119892(119894)120585119894(119883)119875minus1
119892(119894) where 120585
119894(119883) = 119883 or 120585
119894(119883) =
119869119883119905
119869 Let 119875 = diag (1198751 119875
119898) isin T
119898otimesT119899 we complete the
proof of this stepBy Step 3 we may assume
120601 (119864(119898)
119894119894otimes 119883) = 119864
(119898)
119892(119894)119892(119894)otimes 120585119894(119883) forall119894 isin [1119898] 119883 isin T
119899
(48)
Step 4 119892 (119894) = 119894 or 119892(119894) = 119898 minus 119894 + 1Proof of Step 4 If 119898 = 2 then this claim is clear For 119898 ge 3we prove that if 119894 lt 119895 lt 119896 then 119892 (119894) lt 119892 (119895) lt 119892 (119896) or 119892 (119894) gt119892 (119895) gt 119892 (119896) Otherwise we assume 119892 (119894) lt 119892 (119896) lt 119892 (119895)
(other cases can be proven by using similar methods) Sincefor any 119909 isin C (119864(119898)
119894119894+ 119909119864(119898)
119894119895) otimes 119868119899and (119864(119898)
119895119895+ 119909119864(119898)
119894119895) otimes 119868119899are
idempotent matrices inT119898otimesT119899 we have by using (48) that
119864(119898)
119892(119894)119892(119894)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899) and 119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119909120601 (119864
(119898)
119894119895otimes 119868119899)
are idempotent matrices in 119879119898otimes 119879119899 This together with
Lemma 8 implies
120601 (119864(119898)
119894119895otimes 119868119899) = 119864(119898)
119892(119894)119892(119895)otimes 119860 for some119860 = 0 isin T
119899 (49)
Similarly
120601 (119864(119898)
119894119896otimes 119868119899) = 119864(119898)
119892(119894)119892(119896)otimes 119861 for some119861 = 0 isin T
119899
120601 (119864(119898)
119895119896otimes 119868119899) = 119864(119898)
119892(119896)119892(119895)otimes 119862 for some119862 = 0 isin T
119899
(50)
By (119864(119898)119895119895
+119864(119898)
119894119895+119864(119898)
119894119896+119864(119898)
119895119896)otimes119868119899being an idempotent matrix
inT119898otimesT119899 we obtain by using (48) (49) and (50) that
119864(119898)
119892(119895)119892(119895)otimes 119868119899+ 119864(119898)
119892(119894)119892(119895)otimes 119860 + 119864
(119898)
119892(119894)119892(119896)otimes 119861 + 119864
(119898)
119892(119896)119892(119895)otimes 119862
(51)
is an idempotent matrix inT119898otimesT119899 that is
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
2
=
[
[
[
[
[
0 119860 119861
0 0 119862
0 0 119868119899
]
]
]
]
]
(52)
This implies 119860 = 0 which is a contradiction Hence wecomplete the proof of Step 4Step 5 For any 119894 lt 119895 120585
119894= 120585119895 and there exists 120582
119894119895= 0 such that
120601 (119864(119898)
119894119895otimes 119883) = 120582
119894119895119864(119898)
119892(119894)119892(119895)otimes 120585119894(119883) forall119894 = 119895 119883 isin T
119899
(53)
Proof of Step 5 We prove the case of 119892 (119894) = 119894 (one can use asimilar method to prove the case of 119892 (119894) = 119898 minus 119894 + 1) Hence
120601 (119864(119898)
119894119894otimes 119864(119899)
119896119896) = 119864(119898)
119894119894otimes 119864(119899)
119896119896 forall119894 isin [1119898] 119896 isin [1 119899]
(54)
Without loss of generality we may assume 119894 = 1 119895 = 2 Sincefor any 119909 isin C (119864(119898)
11+ 119909119864(119898)
12) otimes 119864(119899)
119896119896and (119864(119898)
22+ 119909119864(119898)
12) otimes 119864(119899)
119896119896
are idempotent matrices inT119898otimesT119899 we obtain by (54) that
119864(119898)
11otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) and119864(119898)
22otimes119864(119899)
119896119896+119909120601 (119864
(119898)
12otimes119864(119899)
119896119896) are
idempotentmatrices in119879119898otimes119879119899This together with Lemma 8
implies 120601(119864(119898)12
otimes 119864(119899)
119896119896) = 120582119896119864(119898)
12otimes 119864(119899)
119896119896 where 120582
119896= 0 Let Λ =
diag (1205821 120582
119898) then
120601 (119864(119898)
12otimes 119868119899) =
[
[
0 Λ
0 0
]
]
oplus 0 (55)
Let119876 = [119868119898
minusΛ
0 119868119898
]oplus119868(119898minus2)119899
isin T119898otimesT119899 by (55) one can obtain
120601 ((119864(119898)
11+ 119864(119898)
12) otimes 119868119899) =
[
[
119868119899Λ
0 0
]
]
oplus 0
= 119876 (119864(119898)
11otimes 119868119899)119876minus1
120601 ((119864(119898)
22minus 119864(119898)
12) otimes 119868119899) =
[
[
0 minusΛ
0 119868119899
]
]
oplus 0
= 119876 (119864(119898)
22otimes 119868119899)119876minus1
(56)
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
Submit your manuscripts athttpwwwhindawicom
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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
Complex AnalysisJournal of
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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
Abstract and Applied Analysis 7
Let 1198651= 119864(119898)
11+ 119864(119898)
12 1198652= 119864(119898)
22minus 119864(119898)
12 Then (56) turn into
120601 (119865119894otimes 119868119899) = 119876 (119864
(119898)
119894119894otimes 119868119899)119876minus1
119894 = 1 2 (57)
By (57) using a similar method to Step 3 one can obtain
120601 (119865119894otimes 119883) = 119876 (119864
(119898)
119894119894otimes 120577119894(119883))119876
minus1
forall119883 isin T119899 119894 = 1 2
(58)
where 120577119894(119883) = 119883 or 120577
119894(119883) = 119869119883
119905
119869 Hence
120601 (1198651otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
1205771(119883) 0
0 0
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
1205771(119883) 120577
1(119883)Λ
0 0
]
]
forall119883 isin T119899
120601 (1198652otimes 119883) =
[
[
119868119898
minusΛ
0 119868119898
]
]
[
[
0 0
0 1205772(119883)
]
]
[
[
119868119898
Λ
0 119868119898
]
]
=[
[
0 minusΛ1205772(119883)
0 1205772(119883)
]
]
forall119883 isin T119899
(59)
Thus
120601 (119864(119898)
12otimes 119883) = 120601 (119865
1otimes 119883) minus 120601 (119864
(119898)
11otimes 119883)
=[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
120601 (119864(119898)
12otimes 119883) = 120601 (119864
(119898)
22otimes 119883) minus 120601 (119865
2otimes 119883)
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(60)
This implies
[
[
1205771(119883) minus 120585
1(119883) 120577
1(119883)Λ
0 0
]
]
=[
[
0 Λ1205772(119883)
0 1205852(119883) minus 120577
2(119883)
]
]
(61)
From 1205771(119883)Λ = Λ120577
2(119883) forall119883 isin T
119899 one can easily see that
Λ = 12058212119868119899
= 0 and 1205771(119883) = 120577
2(119883) thus 120585
1(119883) = 120585
2(119883) This
completes the proof of Step 5By Step 5 we may assume 120585 = 120585
119894
Step 6 For119898 ge 3 and 119894 lt 119895 lt 119896 we have 120582119894119895120582119895119896= 120582119894119896
Proof of Step 6 From (119864(119898)
119895119895+ 119864(119898)
119894119895+ 119864(119898)
119895119896+ 119864(119898)
119894119896) otimes 119868119899is an
idempotent matrix inT119898otimesT119899 we have
(119864(119898)
119892(119895)119892(119895)+ 120582119894119895119864(119898)
119892(119894)119892(119895)+ 120582119895119896119864(119898)
119892(119895)119892(119896)+ 120582119894119896119864(119898)
119892(119894)119892(119896)) otimes 119868119899
(62)
is an idempotent matrix inT119898otimesT119899 It follows from 119892 (119894) = 119894
or 119892 (119894) = 119898 minus 119894 + 1 that 120582119894119895120582119895119896= 120582119894119896
When 119892(119894) = 119894 let 119875 = diag (12058212 120582
1119898) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119894119895119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899 (63)
Hence
120601 (119860 otimes 119883) = (119875 otimes 119868119899) (119860 otimes 120585 (119883)) (119875 otimes 119868
119899)minus1
forall119860 isin T119898 119883 isin T
119899
(64)
When 119892(119894) = 119898 minus 119894 + 1 let 119875 = diag (1205821119898 120582
12) then
120601 (119864(119898)
119894119895otimes 119883) = 119875119864
(119898)
119899minus119894+1119899minus119895+1119875minus1
otimes 120585 (119883) forall119894 le 119895 119883 isin T119899
(65)
Thus
120601 (119860 otimes 119883) = (119875 otimes 119868119899) ((119869119860
119879
119869) otimes 120585 (119883)) (119875 otimes 119868119899)minus1
forall119860 isin T119898 119883 isin T
119899
(66)
This completes the proof of the theorem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the referee for hisher careful reading ofthe paper and valuable comments which greatly improved thereadability of the paper Li Yang is supported by VocationalEducation Institute in Heilongjiang Province ldquo12th five-yeardevelopment planrdquo and the Guiding Function and PracticeResearch ofMathematicalModeling in AdvancedMathemat-ics Teaching of New Rise Financial Institutions (Grant noGG0666) Jinli Xu was supported by the National NaturalScience FoundationGrants of China (Grant no 11171294) andthe Natural Science Foundation of Heilongjiang Province ofChina (Grant no A201013)
References
[1] C-K Li and S Pierce ldquoLinear preserver problemsrdquoThe Ameri-can Mathematical Monthly vol 108 no 7 pp 591ndash605 2001
[2] L Molnar Selected Preserver Problems on Algebraic Structuresof Linear Operators and on Function Spaces vol 1895 of LectureNotes in Mathematics Springer Berlin Germany 2007
[3] G H Chan M H Lim and K-K Tan ldquoLinear preservers onmatricesrdquo Linear Algebra and Its Applications vol 93 pp 67ndash80 1987
[4] P Semrl ldquoApplying projective geometry to transformations onrank one idempotentsrdquo Journal of Functional Analysis vol 210no 1 pp 248ndash257 2004
[5] XM Tang C G Cao and X Zhang ldquoModular automorphismspreserving idempotence and Jordan isomorphisms of triangularmatrices over commutative ringsrdquo Linear Algebra and its Appli-cations vol 338 pp 145ndash152 2001
8 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
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 Abstract and Applied Analysis
[6] N Jacobson Lectures in Abstract Algebra vol 2 D van Nos-trand Toronto Canada 1953
[7] T Oikhberg and A M Peralta ldquoAutomatic continuity oforthogonality preservers on a non-commutative 119871119901(120591) spacerdquoJournal of Functional Analysis vol 264 no 8 pp 1848ndash18722013
[8] M A Chebotar W-F Ke P-H Lee and N-C Wong ldquoMap-pings preserving zero productsrdquo Studia Mathematica vol 155no 1 pp 77ndash94 2003
[9] C-W Leung C-K Ng and N-C Wong ldquoLinear orthogonalitypreservers of Hilbert 119862lowast-modules over 119862lowast-algebras with realrank zerordquo Proceedings of the American Mathematical Societyvol 140 no 9 pp 3151ndash3160 2012
[10] I V Konnov and J C Yao ldquoOn the generalized vector vari-ational inequality problemrdquo Journal of Mathematical Analysisand Applications vol 206 no 1 pp 42ndash58 1997
[11] D Huo B Zheng and H Liu ldquoCharacterizations of nonlinearLie derivations of 119861(119883)rdquo Abstract and Applied Analysis vol2013 Article ID 245452 7 pages 2013
[12] H Yao and B Zheng ldquoZero triple product determined matrixalgebrasrdquo Journal of Applied Mathematics vol 2012 Article ID925092 18 pages 2012
[13] J Xu B Zheng and L Yang ldquoThe automorphism group of theLie ring of real skew-symmetric matricesrdquo Abstract and AppliedAnalysis vol 2013 Article ID 638230 8 pages 2013
[14] X Song C Cao and B Zheng ldquoA note on 119896-potence preserverson matrix spaces over complex fieldrdquo Abstract and AppliedAnalysis vol 2013 Article ID 581683 8 pages 2013
[15] A Fosner Z Haung C K Li and N S Sze ldquoLinear preserversand quantum information sciencerdquo Linear and MultilinearAlgebra vol 61 no 10 pp 1377ndash1390 2013
[16] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000
[17] J Xu B Zheng and H Yao ldquoLinear transformations betweenmultipartite quantum systems that map the set of tensorproduct of idempotent matrices into idempotent matrix setrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 182569 7 pages 2013
[18] B D Zheng J L Xu and A Fosner ldquoLinear maps preservingrank of tensor products of matricesrdquo Linear and MultilinearAlgebra In press
[19] J L Xu B D Zheng and C G Cao ldquoA characterization ofcomparable tableaux rdquo Submitted to Journal of CombinatorialTheory A
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
![FREE IDEMPOTENT GENERATED SEMIGROUPS ANDvarg1/DolinkaGouldYang_1.S2.pdfmatrices over a field is idempotent generated (see also [19]). Fountain and Lewin [10] subsumed these results](https://static.fdocuments.us/doc/165x107/5edcb0a3ad6a402d666776b5/free-idempotent-generated-semigroups-and-varg1dolinkagouldyang1s2pdf-matrices.jpg)
