Optimization problems on the rank and inertia of the Hermitian … · 2010. 10. 6. · generalized...

Optimization problems on the rank and inertia of the Hermitianmatrix expression A−BX − (BX)∗ with applications

Yongge Tian

China Economics and Management Academy, Central University of Finance and Economics, Beijing 100081, China

Abstract. We give in this paper some closed-form formulas for the maximal and minimal values of the rankand inertia of the Hermitian matrix expression A − BX ± (BX)∗ with respect to a variable matrix X. As ap-plications, we derive the extremal values of the ranks/inertias of the matrices X and X ± X∗, where X is a(Hermitian) solution to the matrix equation AXB = C, respectively, and give necessary and sufficient condi-tions for the matrix equation AXB = C to have Hermitian, definite and Re-definite solutions. In particular,we derive the extremal ranks/inertias of Hermitian solutions X of the matrix equation AXA∗ = C, as well asthe extremal ranks/inertias of Hermitian solution X of a pair of matrix equations A1XA∗

1 = C1 and A2XA∗2 = C2.

AMS Classifications: 15A09; 15A24; 15B57

Keywords: Moore–Penrose inverse; matrix expression; matrix equation; inertia; rank; equality; inequality; Her-

mitian solution, definite solution; Re-definite solution; Hermitian perturbation

1 Introduction

In a recent paper [63], the present author studied upper and lower bounds of the rank/inertia of thefollowing linear Hermitian matrix expression (matrix function)

A−BXB∗, (1.1)

where A is a given m×m Hermitian matrix, B is a given m×n matrix, X is an n×n variable Hermitianmatrix X, and B∗ denotes the conjugate transpose of B, and obtained a group of closed-form formulasfor the exact upper and lower bounds (maximal and minimal values) of the rank/inertia of p(X) in (1.2)through pure algebraic operations matrices and their generalized inverses of matrices. The closed-formformulas obtained enable us to derive some valuable consequences on nonsingularity and definiteness of(1.1), as well existence of Hermitian solution of the matrix equation BXB∗ = A.

As a continuation, we consider in this paper the optimization problems on the rank/inertia of theHermitian matrix expression

p(X) = A−BX − (BX)∗, (1.2)

where A = A∗ and B are given m × m and m × n matrices, respectively, and X is an n × m variablematrix. This expression is often encountered in solving some matrix equations with symmetric patternsand in the investigation of Hermitian parts of complex matrices.

The problem of maximizing or minimizing the rank or inertia of a matrix is a special topic in optimiza-tion theory. The maximial/minimimal rank/inertia of a matrix expression can be used to characterize:

(I) the maximal/minimal dimensions of the row and column spaces of the matrix expression;

(II) nonsingularity of the matrix expression when it is square;

(III) solvability of the corresponding matrix equation;

(IV) rank/inertia invariance of the matrix expression;

(V) definiteness of the matrix expression when it is Hermitian;

etc. Notice that the domain of p(X) in (1.2) is the continuous set of all n×m matrices, while the objectivefunctions—the rank and inertia of p(X) take values only from a finite set of nonnegative integers. Hence,this kind of continuous-discrete optimization problems cannot be solved by various optimization methodsfor continuous or discrete cases. It has been realized that rank/inertia optimization and completionproblems have deep connections with computational complexity and numerous important algorithmic

E-mail Address: [email protected]

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applications. Except some special cases as in (1.1) and (1.2), solving rank optimization problems (globally)is very difficult. In fact, optimization problems and completion problems on the rank/inertia of a generalmatrix expression were regarded as NP-hard; see, e.g., [14, 15, 16, 21, 24, 25, 36, 42, 47]. Fortunately,closed-form solutions to the rank/inertia optimization problems of A−BXB∗ and A−BX − (BX)∗, asshown in [37, 39, 63, 67] and Section 2 below, can be derived algebraically by using generalized inversesof matrices.

Throughout this paper, Cm×n and CmH stand for the sets of all m × n complex matrices and all

m×m complex Hermitian matrices, respectively. The symbols AT , A∗, r(A), R(A) and N (A) stand forthe transpose, conjugate transpose, rank, range (column space) and null space of a matrix A ∈ Cm×n,respectively; Im denotes the identity matrix of order m; [A, B ] denotes a row block matrix consistingof A and B. We write A > 0 (A > 0) if A is Hermitian positive (nonnegative) definite. Two Hermitianmatrices A and B of the same size are said to satisfy the inequality A > B (A > B) in the Lowner partialordering if A − B is positive (nonnegative) definite. The Moore–Penrose inverse of A ∈ Cm×n, denotedby A†, is defined to be the unique solution X satisfying the four matrix equations

(i) AXA = A, (ii) XAX = X, (iii) (AX)∗ = AX, (iv) (XA)∗ = XA.

If X satisfies (i), it is called a g-inverse of A and is denoted by A−. A matrix X is called a Hermitiang-inverse of A ∈ Cm

H , denoted by A∼, if it satisfies both AXA = A and X = X∗. Further, the symbolsEA and FA stand for the two orthogonal projectors EA = Im − AA† and FA = In − A†A onto thenull spaces N (A∗) = R(A)⊥ and N (A) = R(A∗)⊥, respectively. The ranks of EA and FA are givenby r(EA) = m − r(A) and r(FA) = n − r(A). A well-known property of the Moore–Penrose inverse is(A†)∗ = (A∗)†. In addition, AA† = A†A if A = A∗. We shall repeatedly use them in the latter part ofthis paper. Results on the Moore–Penrose inverse can be found, e.g., in [4, 5, 28].

The Hermitian part of a square matrix A is defined to be H(A) = (A+A∗)/2. A square matrix A is saidto be Re-positive (Re-nonnegative) definite if H(A) > 0 (H(A) > 0), and Re-negative (Re-nonpositive)definite if H(A) < 0 (H(A) 6 0).

As is well known, the eigenvalues of a Hermitian matrix A ∈ CmH are all real, and the inertia of A is

defined to be the tripletIn(A) = { i+(A), i−(A), i0(A) },

where i+(A), i−(A) and i0(A) are the numbers of the positive, negative and zero eigenvalues of A countedwith multiplicities, respectively. The two numbers i+(A) and i−(A) are called the positive and negativeindex of inertia, respectively, and both of which are usually called the partial inertia of A; see, e.g., [2].For a matrix A ∈ Cm

H , we have

r(A) = i+(A) + i−(A), i0(A) = m− r(A). (1.3)

Hence, once i+(A) and i−(A) are both determined, r(A) and i0(A) are both obtained as well.It is obvious that p(X) = 0, p(X) > 0(> 0, < 0, 6 0) in (1.4) correspond to the well-known matrix

equation and inequalities of Lyapunov type

BX + (BX)∗ = A, BX + (BX)∗ < A(6 A, > A, > A).

Some previous work on these kinds of equation and inequality can be found, e.g., in [6, 26, 27, 35, 67, 72].In addition, the Hermitian part of A + BX (see [31]), Re-definite solutions of the matrix equationsAX = B and AXB = C (see, e.g., [11, 71, 73, 74, 75]), Hermitian solution of the consistent matrixequation AXA∗ = B, as well as the Hermitian generalized inverse of a Hermitian matrix (see, e.g., [63])can also be represented in the form of (1.2). When X runs over Cn×m, the p(X) in (1.2) may varywith respect to the choice of X. In such a case, it is would be of interest to know how the rank, range,nullity, inertia of p(X) vary with respect to X. In two recent papers [37, 67], the p(X) was studied,and the maximal and minimal possible ranks of p(X) with respect to X ∈ Cn×m were derived throughgeneralized inverses of matrices and partitioned matrices. This paper aim at deriving the maximal andminimal possible values of the inertias of p(X) with respect to X through the Moore–Penrose generalizedinverse of matrices, and give closed-form expressions of the matrix X such that the extremal values areattained.

In optimization theory, as well as system and control theory, minimizing/maximizing the rank of apartially specified matrix or matrix expression subject to its variable entries is referred to as a rankminimization/maximization problem, and is denoted collectively by RMPs; see [3, 14, 15, 34, 44, 45].Correspondingly, minimizing/maximizing the inertia of a partially specified Hermitian matrix or matrix

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expression subject to its variable entries is referred to as an inertia minimization/maximization problem,and is denoted collectively by IMPs. RMPs/IMPs now are known to be NP-hard in general, and asatisfactory characterization of the solution set of a general RMP/IMP is currently not available. For alarge amount of RMPs/IMPs associated with linear matrix equations and linear matrix expressions, itis, however, possible to give closed-form solutions through some matrix tools, such as, generalized SVDsand generalized inverses of matrices.

Note that the inertia of a Hermitian matrix divides the eigenvalues of the matrix into three sets onthe real line. Hence, the inertia can be used to characterize definiteness of the Hermitian matrix. Thefollowing results are obvious from the definitions of the rank/inertia of a matrix.

Lemma 1.1 Let A ∈ Cm×m, B ∈ Cm×n, and C ∈ CmH . Then,

(a) A is nonsingular if and only if r(A) = m.

(b) B = 0 if and only if r(B) = 0.

(c) C > 0 (C < 0) if and only if i+(C) = m (i−(C) = m).

(d) C > 0 (C 6 0) if and only if i−(C) = 0 (i+(C) = 0).

Because the rank and partial inertia of a (Hermitian) matrix are fine nonnegative integers, the maximaland minimal values of the rank and partial inertia of a (Hermitian) matrix expression with respect to itsvariable components must exist. Combining this fact with Lemma 1.1, we have the following assertions.

Lemma 1.2 Let S be a set consisting of (square) matrices over Cm×n, and let H be a set consisting ofHermitian matrices over Cm

H . Then,

(a) S has a nonsingular matrix if and only if maxX∈S

r(X) = m.

(b) All X ∈ S are nonsingular if and only if minX∈S

r(X) = m.

(c) 0 ∈ S if and only if minX∈S

r(X) = 0.

(d) S = {0} if and only if maxX∈S

r(X) = 0.

(e) All X ∈ S have the same rank if and only if maxX∈S

r(X) = minX∈S

r(X).

(f) H has a matrix X > 0 (X < 0) if and only if maxX∈H

i+(X) = m

(maxX∈H

i−(X) = m

).

(g) All X ∈ H satisfy X > 0 (X < 0) if and only if minX∈H

i+(X) = m

(minX∈H

i−(X) = m

).

(h) H has a matrix X > 0 (X 6 0) if and only if minX∈H

i−(X) = 0(

minX∈H

i+(X) = 0)

.

(i) All X ∈ H satisfy X > 0 (X 6 0) if and only if maxX∈H

i−(X) = 0(

maxX∈H

i+(X) = 0)

.

(j) All X ∈ H have the same positive index of inertia if and only if maxX∈H

i+(X) = minX∈H

i+(X).

(k) All X ∈ H have the same negative index of inertia if and only if maxX∈H

i−(X) = minX∈H

i−(X).

Lemma 1.3 Let S1 and S2 be two sets consisting of (square) matrices over Cm×n, and let H1 and H2

be two sets consisting of Hermitian matrices over CmH . Then,

(a) There exist X1 ∈ S1 and X2 ∈ S2 such that X1 −X2 is nonsingular if and only if

maxX1∈S1, X2∈S2

r( X1 −X2 ) = m.

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(b) X1 −X2 is nonsingular for all X1 ∈ S1 and X2 ∈ S2 if and only if

minX1∈S1, X2∈S2

r( X1 −X2 ) = m.

(c) There exist X1 ∈ S1 and X2 ∈ S2 such that X1 = X2, i.e., S1 ∩ S2 6= ∅, i.e., if and only if

minX1∈S1, X2∈S2

r( X1 −X2 ) = 0.

(d) S1 ⊆ S2 (S1 ⊇ S2) if and only if

maxX1∈S1

minX2∈S2

r( X1 −X2 ) = 0(

maxX2∈S2

minX1∈S1

r(X1 −X2 ) = 0)

.

(e) There exist X1 ∈ H1 and X2 ∈ H2 such that X1 > X2 (X1 < X2) if and only if

maxX1∈H1, X2∈H2

i+(X1 −X2 ) = m

(max

X1∈H1, X2∈H2i−( X1 −X2 ) = m

).

(f) X1 > X2 (X1 < X2) for all X1 ∈ H1 and X2 ∈ H2 if and only if

minX1∈H1, X2∈H2

i+( X1 −X2 ) = m

(min

X1∈H1, X2∈H2i−( X1 −X2 ) = m

).

(g) There exist X1 ∈ H1 and X2 ∈ H2 such that X1 > X2 (X1 6 X2) if and only if

minX1∈H1, X2∈H2

i−( X1 −X2 ) = 0(

minX1∈H1, X2∈H2

i+( X1 −X2 ) = 0)

.

(h) X1 > X2 (X1 6 X2) for all X1 ∈ H1 and X2 ∈ H2 if and only if

maxX1∈H1, X2∈H2

i−( X1 −X2 ) = 0(

maxX1∈H1, X2∈H2

i+( X1 −X2 ) = 0)

.

These three lemmas show that once some closed-form formulas for (extremal) ranks/inertias of Her-mitian matrices are derived, we can use these formulas to characterize equalities and inequalities forHermitian matrices. This basic algebraic method, referred to as the matrix/inertia method, is availablefor studying various matrix expressions that involve generalized inverses of matrices and arbitrary matri-ces. In the past two decades, the present author and his colleagues established many closed-form formulasfor (extremal) ranks/inertias of (Hermitian) matrices, and used them to derive numerous consequencesand applications; see, e.g., [37, 38, 39, 41, 55, 56, 58, 59, 60, 61, 62, 63, 67, 68, 69].

The following are some known results on ranks/inertias of matrices, which are used later in this paper.

Lemma 1.4 ([43]) Let A ∈ Cm×n, B ∈ Cm×k, and C ∈ Cl×n be given. Then,

r[A, B ] = r(A) + r(EAB) = r(B) + r(EBA), (1.4)

r

[AC

]= r(A) + r(CFA) = r(C) + r(AFC), (1.5)

r

[A BC 0

]= r(B) + r(C) + r(EBAFC), (1.6)

r

[±AA∗ B

B∗ 0

]= r[A, B ] + r(B). (1.7)

We shall repeatedly use the following simple results on partial inertias of Hermitian matrices.

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Lemma 1.5 Let A ∈ CmH , B ∈ Cn

H and P ∈ Cm×n. Then,

i±(P ∗AP ) 6 i±(A), (1.8)i±(PAP ∗) = i±(A), if P is nonsingular, (1.9)

i±(λA) ={

i±(A) if λ > 0i∓(A) if λ < 0 , (1.10)

i±

[A 00 B

]= i±(A) + i±(B), (1.11)

i±

[0 P

P ∗ 0

]= r(P ). (1.12)

The two inequalities in (1.8) were first given in [48], see also [41, Lemma 2]. Eq. (1.9) is the well-knownSylvester’s law of inertia, which was first established in 1852 by Sylvester [54] (see, e.g., [22, Theorem4.5.8] and [37, p. 377]). Eq. (1.10) is from the fact that the eigenvalues of λA are the eigenvalues of Amultiplied by λ. Eq. (1.11) is obvious from the definition of inertia, and (1.12) is well known (see, e.g.,[22, 23]).

Lemma 1.6 ([17, 49]) Let A, B ∈ CmH . The following statements are equivalent:

(a) R(A) ∩R(B) = {0}.

(b) r( A + B ) = r(A) + r(B).

(c) i+( A + B ) = i+(A) + i+(B) and i−( A + B ) = i−(A) + i−(B).

Lemma 1.7 Let A ∈ CmH and B ∈ Cm×n, and denote M =

[A BB∗ 0

]. Then,

i±(M) = r(B) + i±(EBAEB). (1.13)

In particular,

(a) If A > 0, then i+(M) = r[A, B ] and i−(M) = r(B).

(b) If A 6 0, then i+(M) = r(B) and i−(M) = r[A, B ].

(c) i±(A) 6 i±(M) 6 i±(A) + r(B).

An alternative form of (1.13) was given in [25, Theorem 2.1], and a direct proof of (1.13) was givenin [52, Theorem 2.3]. Results (a)–(c) follow from (1.8), (1.13) and Lemma 1.1. Some formulas derivedfrom (1.13) are

i±

[A BFP

FP B∗ 0

]= i±

A B 0C 0 P ∗

0 P 0

− r(P ), (1.14)

r

[A BFP

FP B∗ 0

]= r

A B 0C 0 P ∗

0 P 0

− 2r(P ), (1.15)

i±

[EQAEQ EQBB∗EQ D

]= i±

A B QB∗ D 0Q∗ 0 0

− r(Q), (1.16)

r

[EQAEQ EQBB∗EQ D

]= r

A B QB∗ D 0Q∗ 0 0

− 2r(Q). (1.17)

We shall use them to simplify the inertias of block Hermitian matrices involving Moore–Penrose inversesof matrices.

Lemma 1.8 ([52]) Let A ∈ Cm×n, B ∈ Cp×q and C ∈ Cm×q be given. Then,

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(a) The matrix equationAX = C (1.18)

has a solution for X ∈ Cn×q if and only if R(C) ⊆ R(A), or equivalently, AA†C = C. In this case,the general solution to (1.18) can be written in the following parametric form

X = A†C + FAV, (1.19)

where V ∈ Cn×q is arbitrary.

(b) The matrix equationAXB = C (1.20)

has a solution for X ∈ Cn×p if and only if R(C) ⊆ R(A) and R(C∗) ⊆ R(B∗), or equivalently,AA†CB†B = C. In this case, the general solution to (1.20) can be written in the following parametricform

X = A†CB† + FAV1 + V2EB , (1.21)

where V1, V2 ∈ Cn×p are arbitrary.

Lemma 1.9 Let Aj ∈ Cmj×n, Bj ∈ Cp×qj and Cj ∈ Cmj×qj be given, j = 1, 2. Then,

(a) [50] The pair of matrix equations

A1XB1 = C1 and A2XB2 = C2 (1.22)

have a common solution for X ∈ Cn×p if and only if

R(Cj) ⊆ R(Aj), R(C∗j ) ⊆ R(B∗j ), r

C1 0 A1

0 −C2 A2

B1 B2 0

= r

[A1

A2

]+r[B1, B2 ], j = 1, 2. (1.23)

(b) [57] Under (1.23), the general common solution to (1.22) can be written in the following parametricform

X = X0 + FAV1 + V2EB + FA1V3EB2 + FA2V4EB1 , (1.24)

where A =[

A1

A2

], B = [ B1, B2 ], and the four matrices V1, . . . , V4 ∈ Cn×p are arbitrary.

In order to derive explicit formulas for ranks of block matrices, we use the following three types ofelementary block matrix operation (EBMO, for short):

(I) interchange two block rows (columns) in a block matrix;

(II) multiply a block row (column) by a nonsingular matrix from the left-hand (right-hand) side in ablock matrix;

(III) add a block row (column) multiplied by a matrix from the left-hand (right-hand) side to anotherblock row (column).

In order to derive explicit formulas for the inertia of a block Hermitian matrix, we use the following threetypes of elementary block congruence matrix operation (EBCMO, for short) for a block Hermitian matrixwith the same row and column partition:

(IV) interchange ith and jth block rows, while interchange ith and jth block columns in the blockHermitian matrix;

(V) multiply ith block row by a nonsingular matrix P from the left-hand side, while multiply ith blockcolumn by P ∗ from the right-hand side in the block Hermitian matrix;

(VI) add ith block row multiplied by a matrix P from the left-hand side to jth block row, while add ithblock column multiplied by P ∗ from the right-hand side to jth block column in the block Hermitianmatrix.

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The three types of operation are in fact equivalent to some congruence transformation of a Hermitianmatrix A → PAP ∗, where the nonsingular matrix P is from the elementary block matrix operations tothe block rows of A, and P ∗ is from the elementary block matrix operations to the block columns of A.An example of exposition for such congruence operations associated with (1.2) is given by

P

0 X In

X∗ A BIn B∗ 0

P ∗ =

0 0 In

0 A−BX −X∗B∗ 0In 0 0

, P =

In 0 0−B Im −X∗

0 0 In

.

Because P is nonsingular, it is a simple matter to establish by using (1.9), (1.11) and (1.12) the followingequalities

i±

0 X In

X∗ A BIn B∗ 0

= i±

0 0 In

0 A−BX −X∗B∗ 0In 0 0

= n + i±( A−BX −X∗B∗ ).

In fact, this kind of congruence operations for block Hermitian matrices were widely used by some authorsin the investigations of inertias of block Hermitian matrices; see, e.g., [7, 8, 9, 12, 13, 22, 23, 51, 63, 64, 65].

Because EBCMOs don’t change the inertia of a Hermitian matrix, we shall repeatedly use the algebraicEBCMOs to simplify block Hermitian matrices and to establish equalities for their inertias in the followingsections.

2 Extremal values of the rank/inertia of A−BX − (BX)∗

The problem of maximizing/minimizing the ranks of the two matrix expressions A−BX ± (BX)∗ withrespect to a variable matrix X were studied in [37, 67], in which the following results were given.

Lemma 2.1 Let A = ±A∗ ∈ Cm×m and B ∈ Cm×n be given. Then, the maximal and minimal ranks ofA−BX ± (BX)∗ with respect to X ∈ Cn×m are given by

maxX∈Cn×m

r[A−BX ± (BX)∗ ] = min{m, r

[A BB∗ 0

]}, (2.1)

minX∈Cn×m

r[A−BX ± (BX)∗ ] = r

[A BB∗ 0

]− 2r(B). (2.2)

Hence,

(a) There exists an X ∈ Cn×m such that A−BX±(BX)∗ is nonsingular if and only if r

[A BB∗ 0

]> m.

(b) A−BX ± (BX)∗ is nonsingular for all X ∈ Cn×m if and only if r(A) = m and B = 0.

(c) There exists an X ∈ Cn×m such that BX ± (BX)∗ = A if and only if r

[A BB∗ 0

]= 2r(B), or

equivalently, EBAEB = 0. In this case, the general solution of BX ± (BX)∗ = A can be written as

X = B†A− 12B†ABB† + UB∗ + FBV,

where U = ∓U∗ ∈ Cn×n and V ∈ Cn×m are arbitrary.

(d) A−BX ± (BX)∗ = 0 for all X ∈ Cn×m if and only if both A = 0 and B = 0.

(e) r[A−BX ± (BX)∗ ] = r(A) for all X ∈ Cn×m if and only if B = 0.

The expressions of the matrices Xs satisfying (2.1) and (2.2) were also presented in [37, 67]. Theorem2.1(c) was given in [26], see also [53].

We next derive the extremal inertia of the Hermitian matrix expression A− BX − (BX)∗, and givethe corresponding matrices Xs such that inertia of A−BX − (BX)∗ attain the extremal values.

Theorem 2.2 Let p(X) be as given in (1.2). Then,

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(a) The maximal values of the partial inertia of p(X) are given by

maxX∈Cn×m

i±[ p(X) ] = i±

[A BB∗ 0

]= r(B) + i±(EBAEB). (2.3)

Two matrices satisfying (2.3) are given by

X = B†A− 12B†ABB† + (U ∓ In)B∗ + FBV, (2.4)

respectively, where U = −U∗ ∈ Cn×n and V ∈ Cn×m are arbitrary.

(b) The minimal values of the partial inertia of p(X) are given by

minX∈Cn×m

i±[ p(X) ] = i±

[A BB∗ 0

]− r(B) = i±(EBAEB). (2.5)

A matrix X ∈ Cn×m satisfying the two in (2.5) is given by

X = B†A− 12B†ABB† + UB∗ + FBV, (2.6)

where U = −U∗ ∈ Cn×n and V ∈ Cn×m are arbitrary.

Proof Note from Lemma 1.7(c) that

i±[p(X)] 6 i±

[p(X) BB∗ 0

]6 i±[p(X)] + r(B). (2.7)

By (1.13),

i±

[p(X) BB∗ 0

]= r(B) + i±[EBp(X)EB ] = r(B) + i±(EBAEB). (2.8)

Combining (2.7) and (2.8) leads to

i±(EBAEB) 6 i±[p(X)] 6 r(B) + i±(EBAEB), (2.9)

that is, i±(EBAEB) and r(B) + i±(EBAEB) are lower and upper bounds of i±[p(X)], respectively.Substituting (2.4) into p(X) gives

p(X) = A−BB†A−ABB† + BB†ABB† −BUB∗ − (BUB∗)∗ ± 2BB∗

= EBAEB ± 2BB∗.

Note that R(EBAEB) ∩R(BB∗) = {0}. Hence, it follows from Lemma 1.6 that

i±[p(X)] = i±(EBAEB ± 2BB∗) = i±(EBAEB) + i±(±2BB∗) = r(B) + i±(EBAEB).

These two equalities imply that the right-hand side of (2.9) are the maximal values of the partial inertiaof p(X), establishing (a).

Substituting (2.6) into p(X) gives

p(X) = A−BB†A−ABB† + BB†ABB† −BUB∗ − (BUB∗)∗ = EBAEB .

Hence, i±[p(X)] = i±(EBAEB), establishing (b). �

Lemma 2.1 and Theorem 2.2 formulate explicitly the extremal values of the rank/inertia of the Her-mitian matrix expression A−BX − (BX)∗ with respect to the variable matrix X. Hence, we can easilyuse these formulas and the corresponding Xs, as demonstrated in Lemma 2.1(a)–(d), to study variousoptimization problems on ranks/inertias of Hermitian matrix expressions. As described in Lemma 2.1,one of the important applications of the extremal values of the partial inertia of A−BX − (BX)∗ is tocharacterize the four matrix inequalities BX + (BX)∗ > A (< A, > A, 6 A). In a recent paper [70],these inequalities were considered and the following results were obtained.

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Corollary 2.3 Let A ∈ CmH and B ∈ Cm×n be given. Then,

(a) There exists an X ∈ Cn×m such that

BX + (BX)∗ > A (2.10)

if and only ifEBAEB 6 0. (2.11)

In this case, the general solution to (2.10) can be written as

X =12B†[A + (M + BU )(M + BU )∗ ]( 2Im −BB† ) + V B∗ + FBW, (2.12)

where M = (−EBAEB)1/2, and U, W ∈ Cn×m and V = −V ∗ ∈ Cn×n are arbitrary.

(b) There exists an X ∈ Cn×m such that

BX + (BX)∗ > A (2.13)

if and only ifEBAEB 6 0 and r(EBAEB) = r(EB). (2.14)

In this case, the general solution to (2.13) can be written as (2.12), in which U is any matrix suchthat r[ (−EBAEB)1/2 + BU ] = m, and W ∈ Cn×m and V = −V ∗ ∈ Cn×n are arbitrary.

(c) There exists an X ∈ Cn×m such that

BX + (BX)∗ 6 A (2.15)

if and only ifEBAEB > 0 (2.16)

In this case, the general solution to (2.15) can be written in the following parametric form

X =12B†[A− ( M + BU )(M + BU )∗ ]( 2Im −BB† ) + V B∗ + FBW, (2.17)

where M = (EBAEB)1/2, and U, W ∈ Cn×m and V = −V ∗ ∈ Cn×n are arbitrary.

(d) There exists an X ∈ Cn×m such that

BX + (BX)∗ < A (2.18)

if and only ifEBAEB > 0 and r(EBAEB) = r(EB). (2.19)

In this case, the general solution to (2.18) can be written as (2.17), in which U is any matrix suchthat r[ (EBAEB)1/2 + BU ] = m, and W ∈ Cn×m and V = −V ∗ ∈ Cn×n are arbitrary.

Setting A > 0 in Lemma 2.1 and Theorem 2.2, and applying (1.7) and Lemma 1.8(a) leads to thefollowing result.

Corollary 2.4 Let p(X) be as given in (1.2), and assume A > 0. Then,

maxX∈Cn×m

r[ p(X) ] = min {m, r[A, B ] + r(B)} , (2.20)

minX∈Cn×m

r[ p(X) ] = r[A, B ]− r(B), (2.21)

maxX∈Cn×m

i+[ p(X) ] = r[A, B ], (2.22)

minX∈Cn×m

i+[ p(X) ] = r[A, B ]− r(B), (2.23)

maxX∈Cn×m

i−[ p(X) ] = r(B), (2.24)

minX∈Cn×m

i−[ p(X) ] = 0. (2.25)

9

The expressions of the matrices Xs satisfying (2.20)–(2.25) can routinely be derived from the previousresults, and therefore is omitted. The results in the previous theorem and corollaries can be used to derivealgebraic properties of various matrix expressions that can be written in the form of p(X) in (1.2). Forinstance, the Hermitian part of the linear matrix expression A + BX can be written as

(A + A∗)/2 + [ BX + (BX)∗ ]/2;

the Hermitian part of the linear matrix expression A + BX + Y C can be written as

12(A + A∗) +

12[B, C∗ ]

[XY ∗

]+

12[X∗, Y ]

[B∗

C

].

Hence, some formulas for the extremal ranks and partial inertias of the Hermitian parts of A + BX andA + BX + Y C can trivially be derived from Lemma 2.1 and Theorem 2.2. Some previous work on theinertia of Hermitian part of A + BX was given in [31].

Furthermore, the results in Lemma 2.1 and Theorem 2.2 can be used to characterize relations betweenthe following two matrix expressions

p1(X1) = A1 + B1X1 + (B1X1)∗, p2(X2) = A2 + B2X2 + (B2X2)∗, (2.26)

where Aj ∈ CmH and Bj ∈ Cm×nj are given, and Xj ∈ Cnj×m is a variable matrix, j = 1, 2.

Theorem 2.5 Let p1(X1) and p2(X2) be as given in (2.26), and denote

B = [B1, B2 ], M =[

A1 −A2 BB∗ 0

].

Then,

maxX1∈Cn1×m, X2∈Cn2×m

r[ p1(X1)− p2(X2) ] = min{m, r(M) }, (2.27)

minX1∈Cn1×m, X2∈Cn2×m

r[ p1(X1)− p2(X2) ] = r(M)− 2r(B), (2.28)

maxX1∈Cn1×m, X2∈Cn2×m

i±[ p1(X1)− p2(X2) ] = i±(M), (2.29)

minX1∈Cn1×m, X2∈Cn2×m

i±[ p1(X1)− p2(X2) ] = i±(M)− r(B). (2.30)

Hence,

(a) There exist X1 ∈ Cn1×m and X2 ∈ Cn2×m such that p1(X1)− p2(X2) is nonsingular if and only ifr(M) > m.

(b) p1(X1)−p2(X2) is nonsingular for all X1 ∈ Cn1×m and X2 ∈ Cn2×m if and only if r( A1−A2 ) = mand B = 0.

(c) There exist X1 ∈ Cn1×m and X2 ∈ Cn2×m such that p1(X1) = p2(X2) if and only if r(M) = 2r(B).

(d) p1(X1) = p2(X2) for all X1 ∈ Cn1×m and X2 ∈ Cn2×m if and only if A1 = A2 and B = 0.

(e) There exist X1 ∈ Cn1×m and X2 ∈ Cn2×m such that p1(X1) > p2(X2) (p1(X1) < p2(X2)) if andonly if i+(M) = m (i−(M) = m).

(f) p1(X1) > p2(X2) (p1(X1) < p2(X2)) for all X1 ∈ Cn1×m and X2 ∈ Cn2×m if and only if i−(M) = m(i+(M) = m).

(g) There exist X1 ∈ Cn1×m and X2 ∈ Cn2×m such that p1(X1) > p2(X2) (p1(X1) 6 p2(X2)) i−(M) =r(B) (i+(M) = r(B)).

(h) p1(X1) > p2(X2) (p1(X1) 6 p2(X2)) for all X1 ∈ Cn1×m and X2 ∈ Cn2×m if and only if A1−A2 > 0and B = 0 (A1 −A2 6 0 and B = 0.)

10

Proof The difference of p1(X1) and p2(X2) in (2.26) can be written as

p1(X1)− p2(X2) = A1 −A2 + [B1, −B2 ][

X1

X2

]+ [X∗

1 , X∗2 ]

[B∗

1

−B∗2

]. (2.31)

Applying Lemma 2.1 and Theorem 2.2 to this matrix expression leads to (2.27)–(2.30). Results (a)–(h)follow from (2.27)–(2.30) and Lemma1.2. �

The following result was recently shown in [63].

Lemma 2.6 Let A ∈ CmH , B ∈ Cm×n and C ∈ Cm×p be given, and denote N = [ B, C ]. Then,

maxX∈Cn

H, Y ∈CpH

i±( A−BXB∗ − CY C∗ ) = i±

[A NN∗ 0

], (2.32)

maxX∈Cn

H, Y ∈CpH

i±( A−BXB∗ − CY C∗ ) = r[A, N ]− i∓

[A NN∗ 0

]. (2.33)

Combining Theorem 2.2 with Lemma 2.6 leads to the following result.

Theorem 2.7 Let A ∈ CmH , B ∈ Cm×n and C ∈ Cm×p and D ∈ Cm×q be given, and denote

p(X, Y, Z ) = A−BX − (BX)∗ − CY C∗ −DZD∗, N = [B, C, D ]. (2.34)

Then,

maxX∈Cn×m, Y ∈Cp

H, Z∈CqH

i±[ p(X, Y, Z ) ] = i±

[A NN∗ 0

], (2.35)

minX∈Cn×m, Y ∈Cp

H, Z∈CqH

i±[ p(X, Y, Z ) ] = r

[A NB∗ 0

]− r(B)− i∓

[A NN∗ 0

]. (2.36)

If A > 0, then


H, Z∈CqH

i+[ p(X, Y, Z ) ] = r[A, B, C, D ], (2.37)


H, Z∈CqH

i−[ p(X, Y, Z ) ] = r[B, C, D ], (2.38)


H, Z∈CqH

i±[ p(X, Y, Z ) ] = r[A, B, C, D ]− r[B, C, D ], (2.39)


H, Z∈CqH

i±[ p(X, Y, Z ) ] = 0. (2.40)

Proof Applying (2.3) and (2.5) to (2.34) gives

maxX

i±[ p(X, Y, Z ) ] = i±

[A− CY C∗ −DZD∗ B

B∗ 0

], (2.41)

minX

i±[ p(X, Y, Z ) ] = i±

[A− CY C∗ −DZD∗ B

B∗ 0

]− r(B). (2.42)

Note that [A− CY C∗ −DZD∗ B

B∗ 0

]=

[A BB∗ 0

]−

[C0

]Y [C∗, 0 ]−

[D0

]Z[D∗, 0 ]. (2.43)

Applying (2.32) and (2.33) to (2.43) gives

maxY, Z

i±

([A BB∗ 0

]−

[C0

]Y [C∗, 0 ]−

[D0

]Z[D∗, 0 ]

)= i±

[A NN∗ 0

],

minY, Z

i±

([A BB∗ 0

]−

[C0

]Y [C∗, 0 ]−

[D0

]Z[D∗, 0 ]

)= r

[A NB∗ 0

]− i∓

[A NN∗ 0

].

Substituting them into (2.41) and (2.42) produces (2.35) and (2.36), respectively. �

11

Eqs. (2.35) and (2.36) can simplify further if the given matrices in them satisfy some restriction. Forinstance, if R(B) ⊆ R[C, D ], then


H, Z∈CqH

i±[ p(X, Y, Z ) ] = i±

[A NN∗ 0

], (2.44)


H, Z∈CqH

i±[ p(X, Y, Z ) ] = r

[A NB∗ 0

]− r(B)− i∓

[A NN∗ 0

], (2.45)

where N = [C, D ]. We shall use (2.44) and (2.45) in Section 4 to characterize the existence of nonnegativedefinite solution of the matrix equation AXB = C.

In the remaining of this section, we give the extremal values of the rank and partial inertia of A −BX − (BX)∗ subject to a consistent matrix equation CX = D.

Theorem 2.8 Let p(X) be as given in (1.2), and assume the matrix equation CX = D is solvable forX ∈ Cn×m, where C ∈ Cp×n and D ∈ Cp×m are given. Also, denote

M =

A B D∗

B∗ 0 C∗

D C 0

, N =[

BC

].

Then,

maxCX=D

r[ p(X) ] = min{m, r(M)− 2r(C)}, (2.46)

minCX=D

r[ p(X) ] = r(M)− 2r(N), (2.47)

maxCX=D

i±[ p(X) ] = i±(M)− r(C), (2.48)

minCX=D

i±[ p(X) ] = i±(M)− r(N). (2.49)

Hence,

(a) CX = D has a solution X such that p(X) is nonsingular if and only if r(M) > m + 2r(C).

(b) p(X) is nonsingular for all solutions of CX = D if and only if r(M) = 2r(N) + m.

(c) The two equations CX = D and BX + (BX)∗ = A have a common solution if and only if r(M) =2r(N).

(d) Any solution of CX = D satisfying BX + (BX)∗ = A if and only if r(M) = 2r(C).

(e) The rank of p(X) is invariant subject to CX = D if and only if r(M) = 2r(N)+m or R(B) ⊆ R(C).

(f) CX = D has a solution X satisfying p(X) > 0 (< 0) if and only if i+(M) = r(C)+m (i−(M) = r(C) + m) .

(g) p(X) (< 0) for all solutions of CX = D if and only if i+(M) = r(N) + m (i−(M) = r(N) + m) .

(h) CX = D has a solution X satisfying p(X) > 0 (6 0) if and only if i−(M) = r(N) (i+(M) = r(N)) .

(i) Any solution of CX = D satisfying p(X) > 0 (6 0) if and only if i−(M) = r(C) (i+(M) = r(C)) .

(j) i+[ p(X) ] subject to CX = D ⇔ i−[ p(X) ] is invariant subject to CX = D ⇔ R(B) ⊆ R(C).

Proof Note from Lemma 1.8(a) that the general solution of CX = D can be written as X = C†D+FCV,where V ∈ Cn×m is arbitrary. Substituting it into p(X) gives rise to

p(X) = A−BC†D − (BC†D)∗ −BFCV − V ∗(BFC)∗. (2.50)

12

Applying (2.1), (2.2), (2.3) and (2.5) to it gives

maxCX=D

r[ p(X) ] = maxV ∈Cn×m

r[A−BC†D − (BC†D)∗ −BFCV − V ∗(BFC)∗ ]

= min{

m, r

[A−BC†D − (BC†D)∗ BFC

(BFC)∗ 0

]}, (2.51)

minCX=D

r[ p(X) ] = minV ∈Cn×m

r[A−BC†D − (BC†D)∗ −BFCV − V ∗(BFC)∗ ]

= r


(BFC)∗ 0

]− 2r(BFC), (2.52)

maxCX=D

i±[ p(X) ] = maxV ∈Cn×m

i±[A−BC†D − (BC†D)∗ −BFCV − V ∗(BFC)∗ ]

= i±


(BFC)∗ 0

], (2.53)

minCX=D

i±[ p(X) ] = minV

i±[A−BC†D − (BC†D)∗ −BFCV − V ∗(BFC)∗ ]

= i±


(BFC)∗ 0

]− r(BFC). (2.54)

Applying (1.14) and (1.15), and simplifying by CC†D = D and EBCMOs, we obtain

i±


(BFC)∗ 0

]= i±

A−BC†D − (BC†D)∗ B 0B∗ 0 C∗

0 C 0

− r(C)

= i±

A B D∗

B∗ 0 C∗

D C 0

− r(C)

= i±(M)− r(C), (2.55)

r


(BFC)∗ 0

]= r(M)− 2r(C), (2.56)

r(BFC) = r

[BC

]− r(C) = r(N)− r(C). (2.57)

Substituting (2.55), (2.56) and (2.57) into (2.51)–(2.54) yields (2.46)–(2.49). Results (a)–(j) follow from(2.46)–(2.49) and Lemma1.2. �

3 Extremal values of ranks/inertias of Hermitian parts of solu-tions to some matrix equations

As some applications of results in Section 2, we derive in this section the extremal values of the ranksand partial inertias of for the Hermitian parts of solutions of the two equations in (1.18) and (1.20), andgive some direct consequences of these extremal values.

Theorem 3.1 Let A, B ∈ Cm×n be given, and assume the matrix equation AX = B is solvable forX ∈ Cn×n. Then,

(a) The maximal value of the rank of X + X∗ is

maxAX=B

r(X + X∗ ) = min{n, 2n + r( AB∗ + BA∗ )− 2r(A)}. (3.1)

(b) The minimal value of the rank of X + X∗ is

minAX=B

r(X + X∗ ) = r(AB∗ + BA∗ ). (3.2)

A matrix X ∈ Cn×n satisfying (3.2) is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ + FAUFA, (3.3)

where U = −U∗ ∈ Cn×n is arbitrary.

13

(c) The maximal values of the partial inertia of X + X∗ are

maxAX=B

i±( X + X∗ ) = n + i±( AB∗ + BA∗ )− r(A). (3.4)

A matrix X ∈ Cn×n satisfying the two formulas in (3.4) is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ ± FA + FAUFA, (3.5)


(d) The minimal values of the partial inertia of X + X∗ are

minAX=B

i±(X + X∗ ) = i±( AB∗ + BA∗ ). (3.6)

A matrix X ∈ Cn×n satisfying the the two formulas in (3.6) is given by (3.3).

In particular,

(e) AX = B has a solution such that X +X∗ is nonsingular if and only if r( AB∗+BA∗ ) > 2r(A)−n.

(f) X + X∗ is nonsingular for all solutions of AX = B if and only if r( AB∗ + BA∗ ) = n.

(g) AX = B has a solution satisfying X +X∗ = 0, i.e., AX = B has a skew-Hermitian solution, if andonly if AB∗ + BA∗ = 0. Such a solution is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ + FAUFA, (3.7)


(h) Any solution of AX = B satisfying X + X∗ = 0 if and only if r( AB∗ + BA∗ ) = 2r(A)− 2n.

(i) The rank of X +X∗ is invariant subject to AX = B if and only if r( AB∗+BA∗ ) = n or r(A) = n.

(j) AX = B has a solution satisfying X + X∗ > 0, i.e., AX = B has a Re-positive definite solution, ifand only if i+( AB∗ + BA∗ ) = r(A). Such a solution is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ + FA + FAUFA, (3.8)


(k) AX = B has a solution X ∈ Cn×n satisfying X + X∗ < 0, i.e., AX = B has a Re-negative definitesolution, if and only if i−( AB∗ + BA∗ ) = r(A). Such a matrix is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ − FA + FAUFA, (3.9)


(l) Any solution of AX = B satisfying X + X∗ > 0 (< 0) if and only if i+( AB∗ + BA∗ ) = n(i−( AB∗ + BA∗ ) = n) .

(m) AX = B has a solution satisfying X+X∗ > 0, i.e., AX = B has a Re-nonnegative definite solution,if and only if AB∗ + BA∗ > 0. Such a matrix is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ + FA(U + W )FA, (3.10)

where U = −U∗ ∈ Cn×n and 0 6 W ∈ Cn×n are arbitrary.

(n) AX = B has a solution X satisfying X + X∗ 6 0, i.e., AX = B has a Re-non-positive definitesolution, if and only if AB∗ + BA∗ 6 0. Such a matrix is given by

X = A†B − (A†B)∗ + A†AB∗(A†)∗ + FA(U −W )FA, (3.11)

where U = −U∗ ∈ Cn×n and 0 6 W ∈ Cn×n are arbitrary.

(o) Any solution of AX = B satisfying X + X∗ > 0 (6 0) if and only if AB∗ + BA∗ > 0 and r(A) = n(AB∗ + BA∗ 6 0 and r(A) = n) .

14

(p) i+( X + X∗ ) is invariant subject to CX = D ⇔ i−( X + X∗ ) is invariant subject to CX = D ⇔r(A) = n.

Proof In fact, setting A = 0 and B = Im, and replacing C and D with A and B in (2.46)–(2.49), weobtain (3.1), (3.2), (3.4) and (3.6). It is easy to verify that (3.3) satisfies AX = B. Substituting (3.3)into X + X∗ gives

X + X∗ = A†B − (A†B)∗ + (A†B)∗ −A†B + A†AB∗(A†)∗ + A†BA†A + FAUFA − FAUFA

= A†(AB∗ + BA∗)(A†)∗. (3.12)

Also, note that

A( X + X∗ )A∗ = AA†( AB∗ + BA∗)(A†)∗A∗ = AB∗ + BA∗. (3.13)

Both (3.12) and (3.13) imply that r( X + X∗ ) = r( AB∗ + BA∗ ), that is, (3.3) satisfies (3.2). It is easyto verify that (3.5) satisfies AX = B. Substituting (3.5) into X + X∗ gives

X + X∗ = A†(AB∗ + BA∗)(A†)∗ ± 2FA. (3.14)

Also, note that R(A†) ∩R(FA) = {0}. Hence, (3.14) implies that

i±( X + X∗ ) = i±(AB∗ + BA∗ ) + i±(±FA) = n + i±( AB∗ + BA∗ )− r(A).

that is, (3.5) satisfies (3.4). It is easy to verify that (3.3) satisfies AX = B and (3.6). Results (e)–(p)follow from (a)–(d) and Lemma 1.2. �

The Re-nonnegative definite solutions of the matrix equation AX = B were considered in [11, 19,73, 74]. Theorem 3.1(h) was partially given in these papers. In addition to the Re-nonnegative definitesolutions, we are also able to derive from (2.46)–(2.49) the solutions of AX = B that satisfies X+X∗ > P(< P, > P, 6 P ).

In what follows, we derive the extremal values of the ranks and partial inertias of the Hermitian partsof solutions of the matrix equation AXB = C.

Theorem 3.2 Let A ∈ Cm×n, B ∈ Cn×p and C ∈ Cm×p be given, and assume that the matrix equationAXB = C is solvable for X ∈ Cn×n. Also, denote

M =

0 C AC∗ 0 B∗

A∗ B 0

, N = [ A∗, B ].

Then,

maxAXB=C

r(X + X∗) = min {n, 2n + r(M)− 2r(A)− 2r(B)} , (3.15)

minAXB=C

r(X + X∗) = r(M)− 2r(N), (3.16)

maxAXB=C

i±(X + X∗) = n + i∓(M)− r(A)− r(B), (3.17)

minAXB=C

i±(X + X∗) = i∓(M)− r(N). (3.18)

Hence,

(a) AXB = C has a solution such that X +X∗ is nonsingular if and only if r(M) > 2r(A)+2r(B)−n.

(b) X + X∗ is nonsingular for all solutions of AXB = C if and only if r(M) = 2r(N) + n.

(c) AXB = C has a solution X ∈ Cn×n satisfying X + X∗ = 0 i.e., AXB = C has a skew-Hermitiansolution, if and only if r(M) = 2r(N).

(d) Any solutions of AXB = C are skew-Hermitian if and only if r(M) = 2r(A) + 2r(B)− 2n.

(e) The rank of X + X∗ subject to AXB = C is invariant if and only if r(M) = 2r(N) + n orr(A) = r(B) = n.

15

(f) AXB = C has a solution satisfying X + X∗ > 0 (X + X∗ < 0), i.e., AXB = C has a Re-positive definite solution (a Re-negative definite solution), if and only if i−(M) = r(A) + r(B)(i+(M) = r(A) + r(B)) .

(g) All solutions of AXB = C satisfy X + X∗ > 0 (X + X∗ < 0) if and only if i−(M) = r(N) + n(i+(M) = r(N) + n) .

(h) AXB = C has a solution X ∈ Cn×n satisfying X + X∗ > 0 (X + X∗ 6 0), i.e., AXB = C has aRe-nonnegative definite solution (a Re-nonpositive definite solution), if and only if i+(M) = r(N)(i−(M) = r(N)) .

(i) All solutions of AXB = C satisfy X +X∗ > 0 (X +X∗ 6 0) if and only if i+(M) = r(A)+r(B)−n(i−(M) = r(A) + r(B)− n) .

(j) i+( X + X∗ ) is invariant subject to AXB = C ⇔ i−( X + X∗ ) is invariant subject to AXB = C⇔ r(A) = r(B) = n.

Proof Note from Lemma 1.8(b) that if AXB = C is consistent, the general expression X + X∗ for thesolution of AXB = C can be written as

X + X∗ = A†CB† + (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗, (3.19)

where V =[

V1

V ∗2

]∈ C2n×n is arbitrary. Applying (2.1), (2.2), (2.3) and (2.5) to (3.22) gives

maxAXB=C

r(X + X∗) = maxV

r(A†CB† + (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗

)= min {n, r(J)} , (3.20)

minAXB=C

r(X + X∗) = minV

r(A†CB† + (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗

)= r|(J)− 2r[FA, EB ], (3.21)

maxAXB=C

i±(X + X∗) = maxV

i±(A†CB† + (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗

)= i±(J), (3.22)

minAXB=C

i±(X + X∗) = minV

i±(A†CB† + (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗

)= i±(J)− r[FA, EB ], (3.23)

where

J =

A†CB† + (A†CB†)∗ FA EB

FA 0 0EB 0 0

.

16

Applying (1.14) and simplifying by AA†CB†B = AA†C = CB†B = C and EBCMOs, we obtain

i±(J) = i±


FA 0 0EB 0 0

= i±

A†CB† + (A†CB†)∗ In In 0 0

In 0 0 A∗ 0In 0 0 0 B0 A 0 0 00 0 B∗ 0 0

− r(A)− r(B)

= i±

0 In In − 1

2 (CB†)∗ − 12A†C

In 0 0 A∗ 0In 0 0 0 B

− 12CB† A 0 0 0

− 12 (A†C)∗ 0 B∗ 0 0

− r(A)− r(B)

= i±

0 In 0 0 0In 0 0 0 00 0 0 −A∗ B0 0 −A 1

2CB†A∗ + 12A(B†)∗C∗ 1

2C0 0 B∗ 1

2C∗ 0

− r(A)− r(B)

= n + i±

0 0 −A∗ B0 −A 0 C0 B∗ C∗ 0

− r(A)− r(B)

= n + i∓

0 C AC∗ 0 B∗

A∗ B 0

− r(A)− r(B), (3.24)

Applying (1.5) and simplifying by EBMOs, we obtain

r[FA, EB ] = r

In In

A 00 B∗

− r(A)− r(B) = r

In 00 −A0 B∗

− r(A)− r(B)

= n + r[A∗, B ]− r(A)− r(B). (3.25)

Adding the two equalities in (3.24) gives

i±


FA 0 0EB 0 0

= 2n + r

0 C AC∗ 0 B∗

A∗ B 0

− 2r(A)− 2r(B), (3.26)

Substituting (3.24), (3.25) and (3.26) into (3.20)–(3.23) yields (3.15)–(3.18). Results (a)–(j) follow from(3.15)–(3.18) and Lemma 1.2. �

The existence of Re-definite solution of the matrix equation AXB = C was considered, e.g., in[11, 71, 73, 74, 75], and some identifying conditions were derived through matrix decompositions andgeneralized inverse of matrices among them. In comparison, Theorem 3.2 shows that the existence ofskew-Hermitian solution and Re-definite solution of the matrix equation AXB = C can be characterizedby some explicit equalities for the rank and partial inertia of a Hermitian block matrix composed by thegiven matrices in the equation.

Theorem 3.3 Let A ∈ Cm×n, B ∈ Cn×p, C ∈ Cm×p and P ∈ CnH be given, and assume that the matrix

equation AXB = C is solvable for X ∈ Cn×n. Also, denote

M =

APA∗ C AC∗ 0 B∗

A∗ B 0

, N = [A∗, B ].

17

Then,

maxAXB=C

r(X + X∗ − P ) = min {n, 2n + r(M)− 2r(A)− 2r(B)} , (3.27)

minAXB=C

r( X + X∗ − P ) = r(M)− 2r(N), (3.28)

maxAXB=C

i±( X + X∗ − P ) = n + i∓(M)− r(A)− r(B), (3.29)

minAXB=C

i±( X + X∗ − P ) = i∓(M)− r(N). (3.30)

Hence,

(a) AXB = C has a solution such that X + X∗ − P is nonsingular if and only if r(M) > 2r(A) +2r(B)− n.

(b) X + X∗ − P is nonsingular for all solutions of AXB = C if and only if r(M) = 2r(N) + n.

(c) AXB = C has a solution X ∈ Cn×n satisfying X + X∗ = P if and only if r(M) = 2r(N).

(d) All solutions of AXB = C satisfy X + X∗ = P if and only if r(M) = 2r(A) + 2r(B)− 2n.

(e) The rank of X + X∗ − P subject to AXB = C is invariant if and only if r(M) = 2r(N) + n orr(A) = r(B) = n.

(f) AXB = C has a solution satisfying X +X∗ > P (X +X∗ < P ) if and only if i+(M) = r(A)+r(B)(i−(M) = r(A) + r(B)) .

(g) All solutions of AXB = C satisfy X + X∗ > P (X + X∗ < P ) if and only if i+(M) = r(N) + n(i−(M) = r(N) + n) .

(h) AXB = C has a solution X ∈ Cn×n satisfying X + X∗ > P (X + X∗ 6 P ) if and only ifi−(M) = r(N) (i+(M) = r(N)) .

(i) All solutions of AXB = C satisfy X+X∗ > P (X+X∗ 6 P ) if and only if i−(M) = r(A)+r(B)−n(i+(M) = r(A) + r(B)− n) .

(j) i+( X + X∗ ) is invariant subject to AXB = C ⇔ i−( X + X∗ ) is invariant subject to AXB = C⇔ r(A) = r(B) = n.

Proof Note from Lemma 1.8(b) that if AXB = C is consistent, the general expression X + X∗−P canbe written as

X + X∗ − P = A†CB† + (A†CB†)∗ − P + [FA, EB ]V + V ∗[FA, EB ]∗, (3.31)

where V ∈ C2n×n ia arbitrary. Applying (2.1), (2.2), (2.3) and (2.5) to (3.31) gives

maxAXB=C

r( X + X∗ − P ) = maxV

r(A†CB† + (A†CB†)∗ − P + [FA, EB ]V + V ∗[FA, EB ]∗

)= min {n, r(J)} , (3.32)

minAXB=C

r( X + X∗ − P ) = minV

r(A†CB† + (A†CB†)∗ − P + [FA, EB ]V + V ∗[FA, EB ]∗

)= r(J)− 2r[FA, EB ], (3.33)

maxAXB=C

i±( X + X∗ − P ) = maxV

i±(A†CB† + (A†CB†)∗ − P + [FA, EB ]V + V ∗[FA, EB ]∗

)= i±r(J), (3.34)

minAXB=C

i±( X + X∗ − P ) = minV

i±(A†CB† + (A†CB†)∗ − P + +[FA, EB ]V + V ∗[FA, EB ]∗

)= i±(J)− r[FA, EB ], (3.35)

where

J =

A†CB† + (A†CB†)∗ − P FA EB

FA 0 0EB 0 0

.

18

Applying (1.14) and simplifying by AA†CB†B = AA†C = CB†B = C and EBCMOs, we obtain

i±(J)

= i±

A†CB† + (A†CB†)∗ − P FA EB

FA 0 0EB 0 0

= i±

A†CB† + (A†CB†)∗ − P In In 0 0

In 0 0 A∗ 0In 0 0 0 B0 A 0 0 00 0 B∗ 0 0

− r(A)− r(B)

= i±

0 In In − 1

2 (CB†)∗ + 14PA∗ − 1

2A†C + 14PB

In 0 0 A∗ 0In 0 0 0 B

− 12CB† + 1

4AP A 0 0 0− 1

2 (A†C)∗ + 14B∗P 0 B∗ 0 0

− r(A)− r(B)

= i±

0 In 0 0 0In 0 0 0 00 0 0 −A∗ B0 0 −A 1

2CB†A∗ + 12A(B†)∗C∗ − 1

2APA∗ 12C − 1

4APB0 0 B∗ 1

2C∗ − 14B∗PA∗ 0

− r(A)− r(B)

= n + i±

0 −A∗ B−A −APA∗ CB∗ C∗ 0

− r(A)− r(B)

= n + i∓

APA∗ C AC∗ 0 B∗

A∗ B 0

− r(A)− r(B)

= n + i∓(M)− r(A)− r(B), (3.36)

Adding the two equalities in (3.24) gives

r(J) = 2n + r(M)− 2r(A)− 2r(B), (3.37)

Substituting (3.36), (3.37) and (3.25) into (3.32)–(3.35) yields (3.27)–(3.30). Results (a)–(j) follow from(3.27)–(3.30) and Lemma 1.2. �

Recalling that the generalized inverse A− of a matrix A is a solution of the matrix equation AXA = A,we apply Theorem 3.2 to AXA = A to produce the following result.

Corollary 3.4 Let A ∈ Cm×m. Then,

minA−

r[A− + (A−)∗ ] = r( A + A∗ ) + 2r(A)− 2r[A∗, A ], (3.38)

minA−

i±[A− + (A−)∗ ] = i±(A + A∗ ) + r(A)− r[A∗, A ]. (3.39)

Hence,

(a) There exists an A− such that A− + (A−)∗ = 0 if and only if r( A + A∗ ) + 2r(A) = 2r[A∗, A ].

(b) There exists an A− such that A− + (A−)∗ > 0 if and only if i+( A + A∗ ) + r(A) = r[A∗, A ].

(c) There exists an A− such that A− + (A−)∗ 6 0 if and only if i−( A + A∗ ) + r(A) = r[A∗, A ].

19

4 Extremal values of ranks/inertias of Hermitian solutions tosome matrix equations

Hermitian solutions and definite solutions of the matrix equations AX = B and AXB = C were consid-ered in the literature, and various results were derived; see, e.g., [30, 33]. In this section, we derive somenew results on Hermitian and definite solutions of AXB = C through the matrix rank/inertia methods.

Theorem 4.1 Let A ∈ Cm×n, B ∈ Cn×p and C ∈ Cm×p be given, and assume that the matrix equationAXB = C is solvable for X ∈ Cn×n. Then,

minAXB=C

r(X −X∗ ) = r

C 0 A0 −C∗ B∗

B A∗ 0

− 2r[A∗, B ]. (4.1)

Hence, the following statements are equivalent:

(a) The matrix equation AXB = C has a Hermitian solution for X.

(b) The pair of matrix equations

AY B = C and B∗Y A∗ = C∗ (4.2)

have a common solution for Y.

(c)

R(C) ⊆ R(A), R(C∗) ⊆ R(B∗), r

C 0 A0 −C∗ B∗

B A∗ 0

= 2r[A∗, B ]. (4.3)

In this case, the general Hermitian solution to AXB = C can be written as

X =12(Y + Y ∗), (4.4)

where Y is the common solution to (4.2), or equivalently,

X =12(Y0 + Y ∗

0 ) + EGU1 + (EGU1)∗ + FAU2FA + EBU3EB , (4.5)

where Y0 is a special common solution to (4.2), G = [ A∗, B ], and three matrices U1 ∈ Cn×n, U2, U3 ∈ CnH

are arbitrary.

Proof Note from (1.21) that the difference X −X∗ for the general solution of AXB = C can be writtenas

X −X∗ = A†CB† − (A†CB†)∗ + FAV1 + V2EB − (FAV1)∗ − (V2EB)∗

= A†CB† − (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗,

where V =[

V1

−V ∗2

]is arbitrary. Applying (1.5) and simplifying by EMBOs, we obtain

r[FA, EB ] = r

In In

A 00 B∗

− r(A)− r(B)

= r

In 00 −A0 B∗

− r(A)− r(B) = n + r[A∗, B ]− r(A)− r(B). (4.6)

20

Applying (2.2) to it and simplifying by (1.15), (4.6), AA†C = CB†B = C and EMBOs, we obtain

minAXB=C

r( X −X∗ ) = minV

r(A†CB† − (A†CB†)∗ + [FA, EB ]V + V ∗[FA, EB ]∗

)= r

A†CB† − (A†CB†)∗ FA EB

FA 0 0EB 0 0

− 2r[FA, EB ]

= r

A†CB† − (A†CB†)∗ In In 0 0

In 0 0 A∗ 0In 0 0 0 B0 A 0 0 00 0 B∗ 0 0

− 2n− 2r[A∗, B ]

= r

0 In 0 0 0In 0 0 A∗ 0In 0 0 0 B

−CB† 0 −A 0 0(A†C)∗ 0 B∗ 0 0

− 2n− 2r[A∗, B ]

= r

0 In 0 0 0In 0 0 0 00 0 0 −A∗ B0 0 −A 0 C0 0 B∗ −C∗ 0

− 2n− 2r[A∗, B ]

= r

C 0 A0 −C∗ B∗

B A∗ 0

− 2r[A∗, B ],

establishing (4.1). Equating the right-hand side of (4.1) to zero leads to the equivalence of (a) and (c).Also, note that if AXB = C has a Hermitian solution X0, then it satisfies A∗X0B

∗ = C∗, that is to say,the pair of equations in (4.2) have a common solution X0. Conversely, if the pair of equations in (4.2)have a common solution, then the matrix X in (4.4) is Hermitian and it also satisfies

AXB =12(AY B + AY ∗B) =

12(C + C) = C.

Thus (4.4) is a Hermitian solution to AXB = C. This fact shows that (a) and (b) are equivalent. Also,note that any Hermitian solution X0 to AXB = C is a common solution to (4.2), and can be written as

X0 =12(X0 + X∗

0 ).

Thus the general solution of AXB = C can really be rewritten as (4.4). The equivalence of (b) and (c)follows from Lemma 1.9(a). Solving (4.2) by Lemma 1.8(b) gives the following common general solution

Y = Y0 + EGV1 + V2EG + FAV3FA + EBV4EB ,

where Y0 is a special solution of the pair, G = [A∗, B ], and V1, . . . , V4 ∈ Cn×n are arbitrary. Substitutingit into (4.4) yields

X =12(Y + Y ∗)

=12(Y0 + Y ∗

0 ) +12EG(V1 + V ∗

2 ) +12(V ∗

1 + V2)EG +12FA(V3 + V ∗

3 )FA +12EB(V4 + V ∗

4 )EB

=12(Y0 + Y ∗

0 ) + EGU1 + (EGU1)∗ + FAU2FA + EBU3EB ,

where U1 ∈ Cn×n, U2, U3 ∈ CnH are arbitrary, establishing (4.5). �

Theorem 4.2 Let A ∈ Cm×n, B ∈ Cn×p, C ∈ Cm×p and P ∈ CnH be given, and assume that the matrix

equation AXB = C has a solution X ∈ CnH. Also, denote

M1 =

APA∗ C AC∗ B∗PB B∗

A∗ B 0

, M2 =

0 C AC∗ 0 B∗

A∗ B 0

,

21

N1 =[

A APA∗ CB∗ C∗ B∗PB

], N2 =

[A 0 CB∗ C∗ 0

].

Then,

maxAXB=C, X∈Cn

H

i±( X − P ) = i∓(M1) + n− r(A)− r(B), (4.7)

minAXB=C, X∈Cn

H

i±( X − P ) = r(N1)− i±(M1), (4.8)

maxAXB=C, X∈Cn

H

i±(X) = i∓(M2) + n− r(A)− r(B), (4.9)

minAXB=C, X∈Cn

H

i±(X) = r(N2)− i±(M2). (4.10)

Hence,

(a) AXB = C has a solution X > P (X < P ) if and only if

i−(M1) = r(A) + r(B) (i+(M1) = r(A) + r(B)).

(b) AXB = C has a solution X > P (X 6 P ) if and only if

i−(M1) = r(N1) (i+(M1) = r(N1)).

(c) AXB = C has a solution X > 0 (X < 0) if and only if

i−(M2) = r(A) + r(B) (i+(M2) = r(A) + r(B)).

(d) All solutions of AXB = C satisfy X > 0 (X < 0) if and only if

i+(M2) = r(N2)− n (i−(M2) = r(N2)− n).

(e) AXB = C has a solution X > 0 (X 6 0) if and only if

i−(M2) = r(N2) (i+(M2) = r(N2)).

(f) All solutions of AXB = C satisfy X > 0 (X 6 0) if and only if

i+(M2) = r(A) + r(B)− n (i−(M2) = r(A) + r(B)− n).

Proof We first show that the set inclusion R(FG) ⊆ R[FA, EB ]. Applying (1.5) to [EG, FA, EB ] and[FA, EB ] and simplifying by EBMOs, we obtain

r[EG, FA, EB ] = r

In In In

G∗ 0 00 A 00 0 B

− r(G)− r(A)− r(B)

= r

In 0 00 −G∗ −G∗

0 A 00 0 B∗

− r(G)− r(A)− r(B)

= r

0 0 A0 B∗ 00 A 00 0 B∗

+ n− r(G)− r(A)− r(B) = n + r(G)− r(A)− r(B).

22

Combining it with (4.6) leads to r[FG, FA, EB ] = r[FA, EB ], i.e., R(FG) ⊆ R[FA, EB ]. In this case,applying (2.44) and (2.45) to (4.5) gives

maxAXB=C, X∈Cn

H

i±( X − P ) = maxU1, U2, U3

i± [X0 − P + EGU1 + (EGU1)∗ + FAU2FA + EBU3EB ]

= i±

X0 − P FA EB

FA 0 0EB 0 0

, (4.11)

minAXB=C,X∈Cn

H

i±( X − P ) = minU1, U2, U3

i± [X0 − P + EGU1 + (EGU1)∗ + FAU2FA + EBU3EB ]

= r

[X0 − P FA EB

EG 0 0

]− i∓

X0 − P FA EB

FA 0 0EB 0 0

− r(EG). (4.12)

Applying (1.14) and simplifying by AX0B = C and EBCMOs, we obtain

i±

X0 − P FA EB

FA 0 0EB 0 0

= i±

X0 − P In In 0 0

In 0 0 A∗ 0In 0 0 0 B0 A 0 0 00 0 B∗ 0 0

− r(A)− r(B)

= i±

0 In In − 1

4X0A∗ + 1

4PA∗ − 14X0B + 1

4PBIn 0 0 A∗ 0In 0 0 0 B

− 14AX0 + 1

4AP A 0 0 0− 1

4B∗X0 + 14B∗P 0 B∗ 0 0

− r(A)− r(B)

= i±

0 In 0 0 0In 0 0 0 00 0 0 −A∗ B0 0 −A 1

2AX0A∗ − 1

2APA∗ 14C − 1

4APB0 0 B∗ 1

4C∗ − 14B∗PA∗ 0

− r(A)− r(B)

= i±

0 −A∗ B−A − 1

2APA∗ 12C

B∗ 12C∗ − 1

2B∗PB

+ n− r(A)− r(B)

= i±

0 −A∗ −B−A −APA∗ −C−B∗ −C∗ −B∗PB

+ n− r(A)− r(B)

= i∓(M1) + n− r(A)− r(B),

23

and applying (1.15) and simplifying by AX0B = C and EBMOs, we obtain

r

[X0 − P FA EB

EG 0 0

]= r

X0 − P In In 0

In 0 0 G0 A 0 00 0 B∗ 0

− r(G)− r(A)− r(B)

= r

0 In In PG−X0GIn 0 0 00 A 0 00 0 B∗ 0

− r(G)− r(A)− r(B)

= r

In 0 00 −A AX0G−APG0 B∗ 0

+ n− r(G)− r(A)− r(B)

= r

[−A AX0A

∗ −APA∗ C −APBB∗ 0 0

]+ 2n− r(G)− r(A)− r(B)

= r

[−A −APA∗ C −APBB∗ C∗ 0

]+ 2n− r(G)− r(A)− r(B)

= r

[A APA∗ CB∗ C∗ B∗PB

]+ 2n− r(G)− r(A)− r(B)

= r(N1) + 2n− r(G)− r(A)− r(B).

Substituting them into (4.11) and (4.12) leads to (4.7) and (4.8), respectively. Setting P = 0 in (4.7)and (4.8) yields (4.9) and (4.10), respectively. Results (a)–(f) follow from (4.7)–(4.10) and Lemma 1.4.�

A special case of the matrix equation AXB = C is the matrix equation AXA∗ = C, which wasstudied by many authors; see, e.g., [1, 18, 20, 33, 38, 41]. Applying Theorems 4.1 and 4.2 to AXA∗ = Cleads to the following result.

Corollary 4.3 Let A ∈ Cm×n, C ∈ CmH and P ∈ Cn

H be given. Then,

(a) The matrix equationAXA∗ = C (4.13)

has a solution X ∈ CnH if and only if R(C) ⊆ R(A). In this case, the general Hermitian solution to

(4.13) can be written asX = A†C(A†)∗ + FAU + (FAU)∗, (4.14)

where U ∈ Cn×n is arbitrary.

(b) Under R(C) ⊆ R(A),

maxAXA∗=C, X∈Cn

H

i±( X − P ) = n + i±( C −APA∗ )− r(A), (4.15)

minAXA∗=C, X∈Cn

H

i±( X − P ) = i±( C −APA∗ ), (4.16)

maxAXA∗=C, X∈Cn

H

i±(X) = n + i±(C)− r(A), (4.17)

minAXA∗=C, X∈Cn

H

i±(X) = i±(C). (4.18)

(c) Under R(C) ⊆ R(A), (4.13) has a solution X > P (X < P ) if and only if i+( C −APA∗ ) = r(A)(i−( C −APA∗ ) = r(A)) .

(d) Under R(C) ⊆ R(A), (4.13) has a solution X > P (X 6 P ) if and only if C > APA∗ (C 6 APA∗) .

(e) Under R(C) ⊆ R(A), (4.13) has a solution X > 0 (X < 0) if and only if C > 0 and r(A) = r(C)(C 6 0 and r(A) = r(C)) .

(f) Under R(C) ⊆ R(A), (4.13) has a solution X > 0 (X 6 0) if and only if C > 0 (C 6 0) .

24

Corollary 4.4 Let A ∈ Cm×m be given. Then,

(a) A has a Hermitian g-inverse if and only if

r( A−A∗ ) = 2r[A, A∗ ]− 2r(A). (4.19)

In this case,

maxA∼

r(A∼) = min{m, r(A + A∗ ) + 2m− 2r(A)}, (4.20)

minA∼

r(A∼) = 2r(A)− r( A + A∗ ), (4.21)

maxA∼

i±(A∼) = i±(A + A∗ ) + m− r(A), (4.22)

minA∼

i±(A∼) = r(A)− i∓( A + A∗ ). (4.23)

(b) A has a nonsingular A∼ if and only if r( A + A∗ ) = 2r(A)−m.

(c) The positive index of inertia of A∼ is invariant ⇔ the negative inertia of A∼ is invariant ⇔r( A + A∗ ) = 2r(A)− n.

(d) There exists an A∼ > 0 ⇔ there exists an A∼ > 0 ⇔ i+(A + A∗ ) = r(A).

(e) There exists an A∼ < 0 ⇔ there exists an A∼ 6 0 ⇔ i−( A + A∗ ) = r(A).

As is well known, one of the fundamental concepts in matrix theory is the partition of a matrix. Manyalgebraic properties of a matrix and is operations can be derived from the submatrices in its partitionsand their operations. In order to reveal more properties of Hermitian solutions to (4.13), we partitionthe unknown Hermitian matrix X in (4.13) into a 2× 2 block form

X =[

X1 X2

X∗2 X3

].

Consequently, (4.13) can be rewritten as

[A1, A2 ][

X1 X2

X∗2 X3

][A∗1A∗2

]= C, (4.24)

where A1 ∈ Cm×n1 , A2 ∈ Cm×n2 , X1 ∈ Cn1H , X2 ∈ Cn1×n2 and X3 ∈ Cn2

H with n1 + n2 = n. In whatfollows, we derive the extremal values of the ranks and partial inertias of the submatrices in a Hermitiansolution to (4.24). Note that X1, X2, X3 can be rewritten as

X1 = P1XP ∗1 , X2 = P1XP ∗2 , X3 = P2XP ∗2 , (4.25)

where P1 = [ In1 , 0 ] and P2 = [ 0, In2 ]. Substituting the general solution in (4.14) into (4.25) yields

X1 = P1X0P∗1 + P1FAV1 + V ∗

1 FAP ∗1 , (4.26)X3 = P2X0P

∗2 + P2FAV2 + V ∗

2 EBP ∗2 , (4.27)

where X0 = A†C(A†)∗, V = [V1, V2 ]. For convenience, we adopt the following notation for the collectionsof the submatrices X1 and X3 in (4.24):

S1 ={

Xj ∈ Cn1H

∣∣∣∣ [A1, A2 ][

X1 X2

X∗2 X3

][A∗1A∗2

]= C

}, (4.28)

S3 ={

X3 ∈ Cn2H

∣∣∣∣ [A1, A2 ][

X1 X2

X∗2 X3

][A∗1A∗2

]= C

}. (4.29)

Theorem 4.5 Suppose that the matrix equation (4.24) is consistent. Then,

25

(a) [38] The maximal and minimal values of the ranks of the submatrices X1 and X3 in (4.24) are givenby

maxX1∈S1

r(X1) = min{

n1, r

[C A2

A∗2 0

]− 2r(A) + 2n1

}, (4.30)

minX1∈S1

r(X1) = r

[C A2

A∗2 0

]− 2r(A2), (4.31)

maxX3∈S3

r(X3) = min{

n2, r

[C A1

A∗1 0

]− 2r(A) + 2n2

}, (4.32)

minX3∈S3

r(X3) = r

[C A1

A∗1 0

]− 2r(A1). (4.33)

(b) The extremal partial inertias of the submatrices X1 and X3 in (4.24) are given by

maxX1∈S1

i±(X1) = i±

[C A2

A∗2 0

]− r(A) + n1, (4.34)

minX1∈S1

i±(X1) = i±

[C A2

A∗2 0

]− r(A2), (4.35)

maxX3∈S3

i±(X3) = i±

[C A1

A∗1 0

]− r(A) + n2, (4.36)

minX3∈S3

i±(X3) = i±

[C A1

A∗1 0

]− r(A1). (4.37)

Hence,

(c) Eq. (4.24) has a solution in which X1 is nonsingular if and only if r

[C A2

A∗2 0

]> 2r(A)− n1.

(d) The submatrix X1 in any solution to (4.24) is nonsingular if and only if r

[C A2

A∗2 0

]= 2r(A2)+n1.

(e) Eq. (4.24) has a solution in which X1 = 0 if and only if r

[C A2

A∗2 0

]= 2r(A2).

(f) The submatrix X1 in any solution to (4.24) satisfies X1 = 0 if and only if r

[C A2

A∗2 0

]= 2r(A)−2n1.

(g) Eq. (4.24) has a solution in which X1 > 0 (X1 < 0) if and only if

i+

[C A2

A∗2 0

]= r(A)

(i−

[C A2

A∗2 0

]= r(A)

).

(h) The submatrix X1 in any solution to (4.24) satisfies X1 > 0 (X1 < 0) if and only if

i+

[C A2

A∗2 0

]= n1 + r(A2)

(i−

[C A2

A∗2 0

]= n1 + r(A2)

).

(i) Eq. (4.24) has a solution satisfying X1 > 0 (X1 6 0) if and only if

i−

[C A2

A∗2 0

]= r(A2)

(i+

[C A2

A∗2 0

]= r(A2)

).

(j) The submatrix X1 in any solution to (4.24) satisfies X1 > 0 (X1 6 0) if and only if

i−

[C A2

A∗2 0

]= r(A)− n1

(i−

[C A2

A∗2 0

]= r(A)− n1

).

(k) The positive signature of X1 in (4.24) is invariant ⇔ the negative signature of X1 (4.24) is invariant⇔ R(A1) ∩R(A2) = {0} and r(A1) = n1.

26

Proof Applying (2.3) and (2.5) to (4.26) gives the following expressions

maxX1∈S1

r(X1) = maxV1

r( P1X0P∗1 + P1FAV1 + V ∗

1 FAP ∗1 ) ={

n1, r

[P1X0P

∗1 P1FA

FAP ∗1 0

]}, (4.38)

minX1∈S1

r(X1) = minV1

r( P1X0P∗1 + P1FAV1 + V ∗

1 FAP ∗1 ) = r

[P1X0P

∗1 P1FA

FAP ∗1 0

]− 2r(P1FA), (4.39)

maxX1∈S1

i±(X1) = maxV1

i±( P1X0P∗1 + P1FAV1 + V ∗

1 FAP ∗1 ) = i±

[P1X0P

∗1 P1FA

FAP ∗1 0

], (4.40)

minX1∈S1

i±(X1) = minV1

i±( P1X0P∗1 + P1FAV1 + V ∗

1 FAP ∗1 ) = i±

[P1X0P

∗1 P1FA

FAP ∗1 0

]− r(P1FA). (4.41)

Applying (1.6), (1.14) and simplifying by AX0A∗ = C and EBCMOs, we obtain

i±

[P1X0P

∗1 P1FA

FAP ∗1 0

]= i±

P1X0P∗1 0 P1

0 0 AP ∗1 A∗ 0

− r(A)

= i±

0 − 12P1X0A

∗ P1

− 12AX0P

∗1 0 A

P ∗1 A∗ 0

− r(A)

= i±

0 0 P1

0 C AP ∗1 A∗ 0

− r(A)

= i±

[C A2

A∗2 0

]+ n1 − r(A), (4.42)

r(P1FA) = r

[P1

A

]− r(A) = n1 + r(A2)− r(A). (4.43)

Add the two in (4.42) gives

r

[P1X0P

∗1 P1FA

FAP ∗1 0

]= r

[C A2

A∗2 0

]+ 2n1 − 2r(A). (4.44)

Substituting (4.42), (4.43) and (4.44) into (4.38)–(4.41) produces the desired formulas in (4.30), (4.31),(4.34) and (4.35). Eqs. (4.32), (4.33), (4.36) and (4.37) can be shown similarly. Results (c)–(k) followfrom (4.34), (4.35) and Lemma 1.2. �

5 Common Hermitian solutions of two matrix equations andtheir inertias

Consider a pair of matrix equations

A1X1A∗1 = C1 and A2X2A

∗2 = C2, (5.1)

where Aj ∈ Cmj×n and Cj ∈ Cmj×mj

h are given, j = 1, 2. Applying Lemma 1.9 to the pair, we obtainthe following result.

Theorem 5.1 Assume that the pair of matrix equations in (5.1) are consistent, respectively, and denote

M =

C1 0 A1

0 −C2 A2

A∗1 A∗2 0

. (5.2)

Then,

maxAjXjA∗j =Cj , Xj∈Cn

H, j=1,2r( X1 −X2 ) = {n, r(M) + 2n− 2r(A1)− 2r(A2) } , (5.3)

minAjXjA∗j =Cj , Xj∈Cn

H, j=1,2r( X1 −X2 ) = r(M)− 2r[A∗1, A∗2 ], (5.4)


H, j=1,2i±( X1 −X2 ) = i±(M) + n− r(A1)− r(A2), (5.5)


H, j=1,2i±( X1 −X2 ) = i±(M)− r[A∗1, A∗2 ]. (5.6)

27

(a) There exist two Hermitian solutions X1 and X2 to (5.1) such that X1 −X2 is nonsingular if andonly if r(M) > 2r(A1) + 2r(A2)− n.

(b) The difference X1 −X2 is nonsingular for any two Hermitian solutions X1 and X2 to (5.1) if andonly if r(M) = 2r[A∗1, A∗2 ] + n.

(c) The pair of matrix equations in (5.1) have a common Hermitian solution if and only if r(M) =2r[A∗1, A∗2 ].

(d) The rank of the difference X1−X2 is invariant for any two Hermitian solutions X1 and X2 of (5.1)if and only if r(M) = 2r[A∗1, A∗2 ]− n or r(A1) = r(A2) = n.

(e) There exist two Hermitian solutions X1 and X2 to (5.1) such that X1 > X2 (X1 < X2) if and onlyif i+(M) = r(A1) + r(A2) (i−(M) = r(A1) + r(A2)).

(f) X1 > X2 (X1 < X2) for any two Hermitian solutions X1 and X2 to (5.1) if and only if i+(M) =r[A∗1, A∗2 ] + n (i−(M) = r[A∗1, A∗2 ] + n).

(g) There exist two Hermitian solutions X1 and X2 to (5.1) such that X1 > X2 (X1 6 X2) if and onlyif i−(M) = r[A∗1, A∗2 ] (i+(M) = r[A∗1, A∗2 ]).

(h) X1 > X2 (X1 6 X2) for any pair of Hermitian solutions to (5.1) if and only if i−(M) = r(A1) +r(A2)− n (i+(M) = r(A1) + r(A2)− n).

(i) The positive signature of X1 − X2 is invariant for any pair of solution of (5.1) ⇔ the negativesignature of X1 −X2 is invariant for any pair of solution of (5.1) ⇔ r(A1) = r(A2) = n.

Proof By Corollary 4.3(a), the general Hermitian solutions of the pair of equations in (5.1) can bewritten as

X1 = A†1C1(A†1)∗ + FA1U1 + (FA1U1)∗, X2 = A†2C2(A

†2)∗ − FA2U2 − (FA2U2)∗,

where U1, U2 ∈ Cn×n are arbitrary. Consequently,

X1 −X2 = A†1C1(A†1)∗ −A†2C2(A

†2)∗ + FA1U1 + (FA1U1)∗ + FA2U2 + (FA2U2)∗

= A†1C1(A†1)∗ −A†2C2(A

†2)∗ + [FA1 , FA2 ]U + U∗[FA1 , FA2 ]∗,

where U =[

U1

U2

]is arbitrary. Applying Theorem 2.5 to the difference gives


H, j=1,2r( X1 −X2 )

= maxU∈C2n×n

h

r(A†1C1(A

†1)∗ −A†2C2(A

†2)∗ + [FA1 , FA2 ]U + U∗[FA1 , FA2 ]∗

)= {n, r(J)} , (5.7)


H, j=1,2r( X1 −X2 )

= minU1, U2∈Cn

H

r(A†1C1(A

†1)∗ −A†2C2(A

†2)∗ + [FA1 , FA2 ]U + U∗[FA1 , FA2 ]∗

)= r(J)− 2r[FA1 , FA2 ], (5.8)


H, j=1,2i±( X1 −X2 )

= maxU1, U2∈Cn

H

i±

(A†1C1(A

†1)∗ −A†2C2(A

†2)∗ + [FA1 , FA2 ]U + U∗[FA1 , FA2 ]∗

)= i±(J), (5.9)


H, j=1,2i±( X1 −X2 )

= minU1, U2∈Cn

H

i±

(A†1C1(A

†1)∗ −A†2C2(A

†2)∗ + [FA1 , FA2 ]U + U∗[FA1 , FA2 ]∗

)= i±(J)− r[FA1 , FA2 ], (5.10)

28

where

J =

A†1C1(A†1)∗ −A†2C2(A

†2)∗ FA1 FA2

FA1 0 0FA2 0 0

.

Applying (1.5) and (1.14), and simplifying by A1A†1C1 = C1, A2A

†2C2 = C2, EBMOs and EBCMOs, we

obtain

r[FA1 , FA2 ] = n + r[A∗1, A∗2 ]− r(A1)− r(A2), (5.11)

i±(J) = i±

A†1C1(A†1)∗ −A†2C2(A

†2)∗ FA1 FA2

FA1 0 0FA2 0 0

= i±

A†1C1(A

†1)∗ −A†2C2(A

†2)∗ In In 0 0

In 0 0 A∗1 0In 0 0 0 A∗20 A1 0 0 00 0 A2 0 0

− r(A1)− r(A2)

= i±

0 In In − 1

2A†1C112A†2C2

In 0 0 A∗1 0In 0 0 0 A∗2

− 12C1(A

†1)∗ A1 0 0 0

12C2(A

†2)∗ 0 A2 0 0

− r(A1)− r(A2)

= i±

0 In 0 0 0In 0 0 0 00 0 0 −A∗ A∗20 0 −A1 C1 00 0 A2 0 −C2

− r(A1)− r(A2)

= i±

0 A∗1 A∗2A1 C1 0A2 0 −C2

+ n− r(A1)− r(A2)

= i±(M) + n− r(A1)− r(A2). (5.12)

Adding the two equalities in (5.12) leads to

r(J) = r(M) + 2n− 2r(A1)− 2r(A2). (5.13)

Substituting (5.11), (5.12) and (5.13) into (5.7)–(5.10) yields (5.3)–(5.6). Results (a)–(i) are straightfor-ward from (5.3)–(5.6) and Lemma 1.2. �

We next consider common Hermitian solutions of the pair of equation in (5.1).

Theorem 5.2 Denote

M =

C1 0 A1

0 C2 A2

A∗1 A∗2 0

, N = r

[C1 0 A1

0 C2 A2

].

Then,

(a) The pair of matrix equations in (5.1) have a common solution X ∈ CnH if and only if

R(Cj) ⊆ R(Aj) and r

C1 0 A1

0 −C2 A2

A∗1 A∗2 0

= 2r

[A1

A2

], j = 1, 2. (5.14)

In this case, the general common Hermitian solution to (5.1) can be written as

X = X0 + FAU1 + (FAU1)∗ + FA1U2FA2 + (FA1U2FA2)∗, (5.15)

where X0 is a special solutions to (5.1), A =[

A1

A2

], and U1 ∈ Cn×n and U2, U3 ∈ Cn

H are arbitrary.

29

Proof If the pair of equations in (5.1) have a common Hermitian solution, then (5.14) follows fromTheorem 5.1(c). Conversely, if (5.14) holds, then also by Theorem 5.1(c) there exists a matrix X0

such that A1X0A∗1 = C1 and A2X0A

∗2 = C2. Taking conjugate transpose gives A1X

∗0A∗1 = C1 and

A2X∗0A∗2 = C2. Adding them leads to

A1(X0 + X∗0 )A∗1 = 2C1, A2(X0 + X∗

0 )A∗2 = 2C2,

which mean that the Hermitian matrix (X0 + X∗0 )/2 is also a common solution to (5.1). By Lemma

1.8(b), the general common solution to (5.1) can be expressed as

X = X0 + FAV1 + V2FA + FA1V3FA2 + FA2V4FA1 ,

where X0 is a special solution to (5.1),and V1, . . . , V4 are arbitrary. Correspondingly (X + X∗)/2 can beexpressed as

( X+X∗ )/2 = ( X0+X∗0 )/2+FA( V1+V ∗

2 )/2+( V1+V ∗2 )FA/2+FA1( V3+V ∗

4 )FA2/2+FA2(V ∗3 +V4 )FA1/2,

establishing (5.15). �

In order to derive more algebraic properties of common Hermitian solutions of (5.1), we need to knowthe extremal ranks/inertias of (5.15). We shall present the corresponding results in another paper.

6 The ranks/inertias of some general Hermitian matrix expres-sions

In this section, we derive extremal ranks/inertias of the following two Hermitian matrix expressions

p(X, Y ) = A−BX − (BX)∗ − CY C∗ and p(X, Y, Z) = A−BX − (BX)∗ − CY C∗ −DZD∗.

where A ∈ CmH , B ∈ Cm×n, C ∈ Cm×p and C ∈ Cm×qbe given.

In a recent paper [63], the present author gave the extremal values of the rank/inertia of (1.1) asfollows.

Lemma 6.1 ([63]) Let A ∈ CmH and B ∈ Cm×n be given. Then,

maxX∈Cn

H

r(A−BXB∗ ) = r[A, B ], (6.1)

minX∈Cn

H

r(A−BXB∗ ) = 2r[A, B ]− r

[A BB∗ 0

], (6.2)

maxX∈Cn

H

i±(A−BXB∗ ) = i±

[A BB∗ 0

], (6.3)

minX∈Cn

H

i±(A−BXB∗ ) = r[A, B ]− i∓

[A BB∗ 0

]. (6.4)

Combining Lemma 6.1 with Theorem 2.2, we obtain the following result.

Theorem 6.2 Let A ∈ CmH , B ∈ Cm×n and C ∈ Cm×p be given, and denote

M =

A B CB∗ 0 0C∗ 0 0

, N =[

A B CB∗ 0 0

]. (6.5)

Then,


H

r[ p(X, Y ) ] = min{m, r(N) }, (6.6)


H

r[ p(X, Y ) ] = 2r(N)− r(M)− 2r(B), (6.7)


H

i±[ p(X, Y ) ] = i±(M), (6.8)


H

i±[ p(X, Y ) ] = r(N)− i∓(M)− r(B). (6.9)

Hence,

30

(a) There exist X ∈ Cn×m and Y ∈ CpH such that p(X, Y ) = m if and only if r(N) > m.

(b) p(X, Y ) = m for all X ∈ Cn×m and Y ∈ CpH if and only if r(A) = m and r(M) = 2r(N)−2r(B)−m.

(c) There exist X ∈ Cn×m and Y ∈ CpH such that p(X, Y ) = 0 if and only if r(M) = 2r(N)− 2r(B).

(d) The rank of p(X, Y ) is invariant for all X ∈ Cn×m and Y ∈ CpH if and only if r(M) = 2r(N) −

2r(B)−m, or B = 0 and r

[A CC∗ 0

]= r[A, C ].

(e) There exist X ∈ Cn×m and Y ∈ CpH such that p(X, Y ) > 0 (p(X, Y ) < 0) if and only if i+(M) = m

(i−(M) = m.)

(f) p(X, Y ) > 0 (p(X, Y ) < 0) for all X ∈ Cn×m and Y ∈ CpH if and only if A > 0 (A < 0) and

[B, C ] = 0.

(g) There exist X ∈ Cn×m and Y ∈ CpH such that p(X, Y ) > 0 (p(X, Y ) 6 0) if and only if i+(M) =

r(N)− r(B) (i−(M) = r(N)− r(B)).

(h) p(X, Y ) > 0 (p(X, Y ) 6 0) for all X ∈ Cn×m and Y ∈ CpH if and only if A > 0 (A 6 0) and

[B, C ] = 0.

(i) i+[ p(X, Y ) ] is invariant for all X ∈ Cn×m and Y ∈ CpH ⇔ i−[ p(X, Y ) ] is invariant for any

X ∈ Cn×m and Y ∈ CpH ⇔ B = 0 and r

[A CC∗ 0

]= r[A, C ].

Proof Applying (2.1)–(2.3) and (2.5) to the the variable matrix X of p(X, Y ) in (6.5), we first obtain

maxX∈Cn×m

r[ p(X, Y ) ] = min{m, r

[A− CY C∗ B

B∗ 0

]}, (6.10)

minX∈Cn×m

r[ p(X, Y ) ] = r

[A− CY C∗ B

B∗ 0

]− 2r(B), (6.11)

maxX∈Cn×m

i±[ p(X, Y ) ] = i±

[A− CY C∗ B

B∗ 0

], (6.12)

minX∈Cn×m

i±[ p(X, Y ) ] = i±

[A− CY C∗ B

B∗ 0

]− r(B). (6.13)

Note that [A− CY C∗ B

B∗ 0

]=

[A BB∗ 0

]−

[C0

]Y [C∗, 0 ].

Applying (6.1)–(6.4) gives

maxY ∈Cp

H

r

[A− CY C∗ B

B∗ 0

]= r(N), (6.14)

minY ∈Cp

H

r

[A− CY C∗ B

B∗ 0

]= 2r(N)− r(M), (6.15)

maxY ∈Cp

H

i±

[A− CY C∗ B

B∗ 0

]= i±(M), (6.16)

minY ∈Cp

H

i±

[A− CY C∗ B

B∗ 0

]= r(N)− i∓(M). (6.17)

Substituting (6.14)–(6.17) into (6.10)–(6.13) gives (6.6)–(6.9). Results (a)–(i) follow from (6.6)–(6.9) andLemma 1.2. �

31

Corollary 6.3 Let 0 6 A ∈ CmH , B ∈ Cm×n and C ∈ Cm×p be given, and let p(X, Y ) be as given in

(6.5). Then,


H

r[ p(X, Y ) ] = min{m, r[A, B, C ] + r(B) }, (6.18)


H

r[ p(X, Y ) ] = r[A, B, C ]− r[B, C ], (6.19)


H

i+[ p(X, Y ) ] = r[A, B, C ], (6.20)


H

i−[ p(X, Y ) ] = r[B, C ], (6.21)


H

i+[ p(X, Y ) ] = r[A, B, C ]− r[B, C ], (6.22)


H

i+[ p(X, Y ) ] = 0. (6.23)

Proof Under A > 0, the following equalities

r(M) = r[A, B, C ] + r[B, C ], r(M) = r[A, B, C ] + r(B),i+(M) = r[A, B, C ], i−(M) = r[B, C ]

follow from (1.7) and Lemma 1.7(a). Thus, (6.6)–(6.9) reduce to (6.19)–(6.23). �

The matrix expression in (6.5) may occur in Hermitian solutions of certain matrix equations, suchas, AXA∗ + BY B∗ = C. Hence, Theorem 6.2 can be used to characterize properties of the Hermitiansolutions.

7 Concluding remarks

We derived a variety of closed-form formulas for the (extremal) ranks/inertias of certain Hermitian matrixexpressions, and presented many consequences of these new formulas in characterizing consistency ofmatrix equations, feasibility of matrix inequalities, existence of some special types of solutions of matrixequations, etc., which bring us much new and deep insight into Hermitian matrices and their properties.This work also demonstrates the usefulness of ranks/inertias of matrices in matrix theory and applications.Notice that the derivations in the previous sections are based on some conventional algebraic operationsof matrices and their generalized inverses, while the results obtained are given in various simple andexplicit forms. Hence, it is quite easy to use these results to approach various problems on Hermitianmatrices and their applications.

Because ranks/inertias of (Hermitian) matrices are two of the most basic numerical characteristics,any progress on ranks/inertias will bring some essential development of matrix theory and applications.Motivated by the work in this paper, we propose some problems related to ranks/inertias for furtherresearch:

(a) Consider optimization problems on the rank/inertia of A − BXB∗ subject to PXQ = N andX = X∗.

(b) Consider existence of solutions of the consistent matrix equations AX = B, AXA∗ = B andAXB = C with some quadratic restrictions, such as X∗X = In, X∗PX > Q, etc.

(c) Recall that the least-squares solution of the matrix equation AXB = C is in fact the solution of itsnormal equation

A∗AXBB∗ = A∗CB∗.

This equation is always consistent. Hence, it would be interest to derive the extremal values of theranks/inertias of the X + X∗ for the least-squares solution of AXB = C, as well as the existenceof Hermitian least-squares solutions that satisfy X > P (X 6 P ) through the matrix rank/inertiamethod.

(d) Note that any Hermitian matrix can be written as A = A0 + jA1, where both A0 and A1 are realmatrices that satisfy AT

0 = A0 and AT1 = −A1, and j =

√−1. In this case, determine the extremal

values of the inertias of A with respect to the skew-symmetric matrix A1.

32

(e) Determine the extremal values of the inertias of A with respect to the real symmetric matrix X0

and skew-symmetric matrix X1 in Hermitian solutions of the consistent matrix equation A(X0 +iX1 )B = C.

(f) Assume the each of the three matrix equations A1X1A∗1 = B1, A2X2A

∗2 = B2, and A3X3A

∗3 = B3

has a solution. Then consider the following equalities and inequalities for their Hermitian solutions

X1 = X2 + X3, X1 > (>, <, 6)X2 + X3.

Some recent work on this problem can be found in [66].

(g) A more challenging task is to derive extremal ranks/inertias of some quadratical matrix expressions,such as,

A−BX − (BX)∗ −X∗CX, E − (A−BXC )D( A−BXC )∗,

A−XBX∗ s.t. PX + (PX)∗ = Q.

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