Optimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications

Optimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications Yongge Tian China Economics and Management Academy, Central University of Finance and Economics, Beijing 181, China Abstract. We give in this paper some closed-form formulas for the maximal and minimal values of the rank and inertia of the Hermitian matrix expression A BX ± (BX) with respect to a variable matrix X. As applications, 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 conditions 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 as the extremal ranks/inertias of Hermitian solution X of a pair of matrix equations A 1XA 1 = C 1 and A 2XA 2 = C 2. AMS Classifications: 15A9; 15A24; 15B57 Keywords: Moore Penrose inverse; matrix expression; matrix equation; inertia; rank; equality; inequality; Hermitian 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 the following 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 Hermitian matrix X, and B denotes the conjugate transpose of B, and obtained a group of closed-form formulas for 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-form formulas 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 the Hermitian 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 variable matrix. This expression is often encountered in solving some matrix equations with symmetric patterns and 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 optimization 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 objective functions 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 methods for continuous or discrete cases. It has been realized that rank/inertia optimization and completion problems have deep connections with computational complexity and numerous important algorithmic E-mail Address: yongge.tian@gmail.com 1

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 general matrix 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), as shown in [37, 39, 63, 67 and Section 2 below, can be derived algebraically by using generalized inverses of matrices. Throughout this paper, C m n and C m H stand for the sets of all m n complex matrices and all m m complex Hermitian matrices, respectively. The symbols A T, A, r(a), R(A) and N (A) stand for the transpose, conjugate transpose, rank, range (column space) and null space of a matrix A C m n, respectively; I m denotes the identity matrix of order m; [ A, B denotes a row block matrix consisting of A and B. We write A > (A ) if A is Hermitian positive (nonnegative) definite. Two Hermitian matrices A and B of the same size are said to satisfy the inequality A > B (A B) in the Löwner partial ordering if A B is positive (nonnegative) definite. The Moore Penrose inverse of A C m n, denoted by 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 Hermitian g-inverse of A C m H, denoted by A, if it satisfies both AXA = A and X = X. Further, the symbols E A and F A stand for the two orthogonal projectors E A = I m AA and F A = I n A A onto the null spaces N (A ) = R(A) and N (A) = R(A ), respectively. The ranks of E A and F A are given by r(e A ) = m r(a) and r(f A ) = 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 of this 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 said to be Re-positive (Re-nonnegative) definite if H(A) > (H(A) ), and Re-negative (Re-nonpositive) definite if H(A) < (H(A) ). As is well known, the eigenvalues of a Hermitian matrix A C m H are all real, and the inertia of A is defined to be the triplet In(A) = { i + (A), i (A), i (A) }, where i + (A), i (A) and i (A) are the numbers of the positive, negative and zero eigenvalues of A counted with multiplicities, respectively. The two numbers i + (A) and i (A) are called the positive and negative index of inertia, respectively, and both of which are usually called the partial inertia of A; see, e.g., [2. For a matrix A C m H, we have r(a) = i + (A) + i (A), i (A) = m r(a). (1.3) Hence, once i + (A) and i (A) are both determined, r(a) and i (A) are both obtained as well. It is obvious that p(x) =, p(x) > (, <, ) in (1.4) correspond to the well-known matrix equation and inequalities of Lyapunov type BX + (BX) = A, BX + (BX) < A( 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 equations AX = B and AXB = C (see, e.g., [11, 71, 73, 74, 75), Hermitian solution of the consistent matrix equation 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 C n m, the p(x) in (1.2) may vary with 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 C n m were derived through generalized inverses of matrices and partitioned matrices. This paper aim at deriving the maximal and minimal possible values of the inertias of p(x) with respect to X through the Moore Penrose generalized inverse of matrices, and give closed-form expressions of the matrix X such that the extremal values are attained. In optimization theory, as well as system and control theory, minimizing/maximizing the rank of a partially specified matrix or matrix expression subject to its variable entries is referred to as a rank minimization/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 2

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 a satisfactory characterization of the solution set of a general RMP/IMP is currently not available. For a large amount of RMPs/IMPs associated with linear matrix equations and linear matrix expressions, it is, however, possible to give closed-form solutions through some matrix tools, such as, generalized SVDs and generalized inverses of matrices. Note that the inertia of a Hermitian matrix divides the eigenvalues of the matrix into three sets on the real line. Hence, the inertia can be used to characterize definiteness of the Hermitian matrix. The following results are obvious from the definitions of the rank/inertia of a matrix. Lemma 1.1 Let A C m m, B C m n, and C C m H. Then, (a) A is nonsingular if and only if r(a) = m. (b) B = if and only if r(b) =. (c) C > (C < ) if and only if i + (C) = m (i (C) = m). (d) C (C ) if and only if i (C) = (i + (C) = ). Because the rank and partial inertia of a (Hermitian) matrix are fine nonnegative integers, the maximal and minimal values of the rank and partial inertia of a (Hermitian) matrix expression with respect to its variable 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 C m n, and let H be a set consisting of Hermitian matrices over C m H. Then, (a) S has a nonsingular matrix if and only if max r(x) = m. X S (b) All X S are nonsingular if and only if min r(x) = m. X S (c) S if and only if min r(x) =. X S (d) S = {} if and only if max r(x) =. X S (e) All X S have the same rank if and only if max r(x) = min r(x). X S X S (f) H has a matrix X > (X < ) if and only if max X H i +(X) = m (g) All X H satisfy X > (X < ) if and only if min X H i +(X) = m (h) H has a matrix X (X ) if and only if min X H i (X) = (i) All X H satisfy X (X ) if and only if max X H i (X) = ( ) max i (X) = m. X H ( ) min i (X) = m. X H ( ) min i +(X) =. X H ( ) max i +( X) =. X H (j) All X H have the same positive index of inertia if and only if max X H i +(X) = min X H i +(X). (k) All X H have the same negative index of inertia if and only if max X H i (X) = min X H i (X). Lemma 1.3 Let S 1 and S 2 be two sets consisting of (square) matrices over C m n, and let H 1 and H 2 be two sets consisting of Hermitian matrices over C m H. Then, (a) There exist X 1 S 1 and X 2 S 2 such that X 1 X 2 is nonsingular if and only if max r( X 1 X 2 ) = m. X 1 S 1, X 2 S 2 3

(b) X 1 X 2 is nonsingular for all X 1 S 1 and X 2 S 2 if and only if min r( X 1 X 2 ) = m. X 1 S 1, X 2 S 2 (c) There exist X 1 S 1 and X 2 S 2 such that X 1 = X 2, i.e., S 1 S 2, i.e., if and only if min r( X 1 X 2 ) =. X 1 S 1, X 2 S 2 (d) S 1 S 2 (S 1 S 2 ) if and only if max X 1 S 1 min r( X 1 X 2 ) = X 2 S 2 ( max X 2 S 2 ) min r( X 1 X 2 ) =. X 1 S 1 (e) There exist X 1 H 1 and X 2 H 2 such that X 1 > X 2 (X 1 < X 2 ) if and only if max i + ( X 1 X 2 ) = m X 1 H 1, X 2 H 2 ( ) max i ( X 1 X 2 ) = m. X 1 H 1, X 2 H 2 (f) X 1 > X 2 (X 1 < X 2 ) for all X 1 H 1 and X 2 H 2 if and only if min i + ( X 1 X 2 ) = m X 1 H 1, X 2 H 2 ( ) min i ( X 1 X 2 ) = m. X 1 H 1, X 2 H 2 (g) There exist X 1 H 1 and X 2 H 2 such that X 1 X 2 (X 1 X 2 ) if and only if min i ( X 1 X 2 ) = X 1 H 1, X 2 H 2 ( ) min i + ( X 1 X 2 ) =. X 1 H 1, X 2 H 2 (h) X 1 X 2 (X 1 X 2 ) for all X 1 H 1 and X 2 H 2 if and only if max i ( X 1 X 2 ) = X 1 H 1, X 2 H 2 ( ) max i + ( X 1 X 2 ) =. X 1 H 1, X 2 H 2 These three lemmas show that once some closed-form formulas for (extremal) ranks/inertias of Hermitian matrices are derived, we can use these formulas to characterize equalities and inequalities for Hermitian matrices. This basic algebraic method, referred to as the matrix/inertia method, is available for studying various matrix expressions that involve generalized inverses of matrices and arbitrary matrices. In the past two decades, the present author and his colleagues established many closed-form formulas for (extremal) ranks/inertias of (Hermitian) matrices, and used them to derive numerous consequences and applications; see, e.g., [37, 38, 39, 41, 55, 56, 58, 59, 6, 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 C m n, B C m k, and C C l n be given. Then, r[ A, B = r(a) + r(e A B) = r(b) + r(e B A), (1.4) A r = r(a) + r(cf C A ) = r(c) + r(af C ), (1.5) A B r = r(b) + r(c) + r(e C B AF C ), (1.6) ±AA B r B = r[ A, B + r(b). (1.7) We shall repeatedly use the following simple results on partial inertias of Hermitian matrices. 4

Lemma 1.5 Let A C m H, B Cn H and P Cm n. Then, i ± (P AP ) i ± (A), (1.8) i ± (P AP ) (A), if P is nonsingular, (1.9) { i± (A) if λ > i ± (λa) = i (A) if λ <, (1.1) A i ± = i B ± (A) + i ± (B), (1.11) P i ± P = 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-known Sylvester s law of inertia, which was first established in 1852 by Sylvester [54 (see, e.g., [22, Theorem 4.5.8 and [37, p. 377). Eq. (1.1) is from the fact that the eigenvalues of λa are the eigenvalues of A multiplied 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 C m H. The following statements are equivalent: (a) R(A) R(B) = {}. (b) r( A + B ) = r(a) + r(b). (c) i + ( A + B ) = i + (A) + i + (B) and i ( A + B ) = i (A) + i (B). A B Lemma 1.7 Let A C m H and B Cm n, and denote M = B. Then, In particular, (a) If A, then i + (M) = r[ A, B and i (M) = r(b). (b) If A, then i + (M) = r(b) and i (M) = r[ A, B. (c) i ± (A) i ± (M) i ± (A) + r(b). i ± (M) = r(b) + i ± (E B AE B ). (1.13) An alternative form of (1.13) was given in [25, Theorem 2.1, and a direct proof of (1.13) was given in [52, Theorem 2.3. Results (a) (c) follow from (1.8), (1.13) and Lemma 1.1. Some formulas derived from (1.13) are [ A BF i P ± F P B = i ± A B C P r(p ), (1.14) P [ A BF r P F P B = r A B C P 2r(P ), (1.15) P EQ AE i Q E Q B ± B = i E Q D ± A B Q B D r(q), (1.16) Q EQ AE r Q E Q B B = r A B Q B E Q D D 2r(Q). (1.17) Q We shall use them to simplify the inertias of block Hermitian matrices involving Moore Penrose inverses of matrices. Lemma 1.8 ([52) Let A C m n, B C p q and C C m q be given. Then, 5

(a) The matrix equation AX = C (1.18) has a solution for X C n 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 where V C n q is arbitrary. (b) The matrix equation X = A C + F A V, (1.19) AXB = C (1.2) has a solution for X C n 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 (1.2) can be written in the following parametric form X = A CB + F A V 1 + V 2 E B, (1.21) where V 1, V 2 C n p are arbitrary. Lemma 1.9 Let A j C mj n, B j C p qj and C j C mj qj be given, j = 1, 2. Then, (a) [5 The pair of matrix equations A 1 XB 1 = C 1 and A 2 XB 2 = C 2 (1.22) have a common solution for X C n p if and only if C 1 A 1 R(C j ) R(A j ), R(Cj ) R(Bj ), r C 2 A 2 A1 = r[ +r[ B A 1, B 2, j = 1, 2. (1.23) B 1 B 2 2 (b) [57 Under (1.23), the general common solution to (1.22) can be written in the following parametric form X = X + F A V 1 + V 2 E B + F A1 V 3 E B2 + F A2 V 4 E B1, (1.24) A1 where A =, B = [ B A 1, B 2, and the four matrices V 1,..., V 4 C n p are arbitrary. 2 In order to derive explicit formulas for ranks of block matrices, we use the following three types of elementary 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 a block matrix; (III) add a block row (column) multiplied by a matrix from the left-hand (right-hand) side to another block row (column). In order to derive explicit formulas for the inertia of a block Hermitian matrix, we use the following three types of elementary block congruence matrix operation (EBCMO, for short) for a block Hermitian matrix with the same row and column partition: (IV) interchange ith and jth block rows, while interchange ith and jth block columns in the block Hermitian matrix; (V) multiply ith block row by a nonsingular matrix P from the left-hand side, while multiply ith block column 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 ith block column multiplied by P from the right-hand side to jth block column in the block Hermitian matrix. 6

The three types of operation are in fact equivalent to some congruence transformation of a Hermitian matrix A P AP, where the nonsingular matrix P is from the elementary block matrix operations to the 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 X I n X A B I n B P = I n A BX X B I n, P = I n B I m X. I n Because P is nonsingular, it is a simple matter to establish by using (1.9), (1.11) and (1.12) the following equalities i ± X I n X A B I n A BX X B = n + i ± ( A BX X B ). I n B I n In fact, this kind of congruence operations for block Hermitian matrices were widely used by some authors in 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 algebraic EBCMOs to simplify block Hermitian matrices and to establish equalities for their inertias in the following sections. 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) with respect to a variable matrix X were studied in [37, 67, in which the following results were given. Lemma 2.1 Let A = ±A C m m and B C m n be given. Then, the maximal and minimal ranks of A BX ± (BX) with respect to X C n m are given by { } A B max r[ A BX ± X C (BX) = min m, r n m B, (2.1) A B min r[ A BX ± X C (BX) = r n m B 2r(B). (2.2) Hence, A B (a) There exists an X C n m such that A BX±(BX) is nonsingular if and only if r B m. (b) A BX ± (BX) is nonsingular for all X C n m if and only if r(a) = m and B =. A B (c) There exists an X C n m such that BX ± (BX) = A if and only if r B = 2r(B), or equivalently, E B AE B =. In this case, the general solution of BX ± (BX) = A can be written as X = B A 1 2 B ABB + UB + F B V, where U = U C n n and V C n m are arbitrary. (d) A BX ± (BX) = for all X C n m if and only if both A = and B =. (e) r[ A BX ± (BX) = r(a) for all X C n m if and only if B =. The expressions of the matrices Xs satisfying (2.1) and (2.2) were also presented in [37, 67. Theorem 2.1(c) was given in [26, see also [53. We next derive the extremal inertia of the Hermitian matrix expression A BX (BX), and give the 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, 7

(a) The maximal values of the partial inertia of p(x) are given by A B max i ± [ p(x) X C n m B = r(b) + i ± (E B AE B ). (2.3) Two matrices satisfying (2.3) are given by X = B A 1 2 B ABB + (U I n )B + F B V, (2.4) respectively, where U = U C n n and V C n m are arbitrary. (b) The minimal values of the partial inertia of p(x) are given by A B min i ± [ p(x) X C n m B r(b) = i ± (E B AE B ). (2.5) A matrix X C n m satisfying the two in (2.5) is given by where U = U C n n and V C n m are arbitrary. X = B A 1 2 B ABB + UB + F B V, (2.6) Proof Note from Lemma 1.7(c) that p(x) B i ± [p(x) i ± B i ± [p(x) + r(b). (2.7) By (1.13), p(x) B i ± B = r(b) + i ± [ E B p(x)e B = r(b) + i ± (E B AE B ). (2.8) Combining (2.7) and (2.8) leads to i ± (E B AE B ) i ± [p(x) r(b) + i ± (E B AE B ), (2.9) that is, i ± (E B AE B ) and r(b) + i ± (E B AE B ) 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 = E B AE B ± 2BB. Note that R(E B AE B ) R(BB ) = {}. Hence, it follows from Lemma 1.6 that i ± [p(x) (E B AE B ± 2BB ) (E B AE B ) + i ± (±2BB ) = r(b) + i ± (E B AE B ). These two equalities imply that the right-hand side of (2.9) are the maximal values of the partial inertia of p(x), establishing (a). Substituting (2.6) into p(x) gives p(x) = A BB A ABB + BB ABB BUB (BUB ) = E B AE B. Hence, i ± [p(x) (E B AE B ), establishing (b). Lemma 2.1 and Theorem 2.2 formulate explicitly the extremal values of the rank/inertia of the Hermitian matrix expression A BX (BX) with respect to the variable matrix X. Hence, we can easily use these formulas and the corresponding Xs, as demonstrated in Lemma 2.1(a) (d), to study various optimization 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 to characterize the four matrix inequalities BX + (BX) > A (< A, A, A). In a recent paper [7, these inequalities were considered and the following results were obtained. 8

Corollary 2.3 Let A C m H and B Cm n be given. Then, (a) There exists an X C n m such that BX + (BX) A (2.1) if and only if E B AE B. (2.11) In this case, the general solution to (2.1) can be written as X = 1 2 B [ A + ( M + BU )( M + BU ) ( 2I m BB ) + V B + F B W, (2.12) where M = ( E B AE B ) 1/2, and U, W C n m and V = V C n n are arbitrary. (b) There exists an X C n m such that BX + (BX) > A (2.13) if and only if E B AE B and r(e B AE B ) = r(e B ). (2.14) In this case, the general solution to (2.13) can be written as (2.12), in which U is any matrix such that r[ ( E B AE B ) 1/2 + BU = m, and W C n m and V = V C n n are arbitrary. (c) There exists an X C n m such that BX + (BX) A (2.15) if and only if E B AE B (2.16) In this case, the general solution to (2.15) can be written in the following parametric form X = 1 2 B [ A ( M + BU )( M + BU ) ( 2I m BB ) + V B + F B W, (2.17) where M = (E B AE B ) 1/2, and U, W C n m and V = V C n n are arbitrary. (d) There exists an X C n m such that BX + (BX) < A (2.18) if and only if E B AE B and r(e B AE B ) = r(e B ). (2.19) In this case, the general solution to (2.18) can be written as (2.17), in which U is any matrix such that r[ (E B AE B ) 1/2 + BU = m, and W C n m and V = V C n n are arbitrary. Setting A in Lemma 2.1 and Theorem 2.2, and applying (1.7) and Lemma 1.8(a) leads to the following result. Corollary 2.4 Let p(x) be as given in (1.2), and assume A. Then, max r[ p(x) = min {m, r[ A, B + r(b)}, (2.2) X Cn m min r[ p(x) = r[ A, B r(b), (2.21) X Cn m max X C n m i +[ p(x) = r[ A, B, (2.22) min i +[ p(x) = r[ A, B r(b), (2.23) X C n m max X C n m i [ p(x) = r(b), (2.24) min i [ p(x) =. (2.25) X C n m 9

The expressions of the matrices Xs satisfying (2.2) (2.25) can routinely be derived from the previous results, and therefore is omitted. The results in the previous theorem and corollaries can be used to derive algebraic properties of various matrix expressions that can be written in the form of p(x) in (1.2). For instance, 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 1 2 (A + A ) + 1 X 2 [ B, C Y + 1 B 2 [ X, Y. C Hence, some formulas for the extremal ranks and partial inertias of the Hermitian parts of A + BX and A + BX + Y C can trivially be derived from Lemma 2.1 and Theorem 2.2. Some previous work on the inertia 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 between the following two matrix expressions p 1 (X 1 ) = A 1 + B 1 X 1 + (B 1 X 1 ), p 2 (X 2 ) = A 2 + B 2 X 2 + (B 2 X 2 ), (2.26) where A j C m H and B j C m nj are given, and X j C nj m is a variable matrix, j = 1, 2. Theorem 2.5 Let p 1 (X 1 ) and p 2 (X 2 ) be as given in (2.26), and denote Then, Hence, B = [ B 1, B 2, M = A1 A 2 B B. max r[ p 1(X 1 ) p 2 (X 2 ) = min{ m, r(m) }, (2.27) X 1 C n 1 m, X 2 C n 2 m min r[ p 1 (X 1 ) p 2 (X 2 ) = r(m) 2r(B), (2.28) X 1 C n 1 m, X 2 C n 2 m max i ±[ p 1 (X 1 ) p 2 (X 2 ) (M), (2.29) X 1 C n 1 m, X 2 C n 2 m min i ±[ p 1 (X 1 ) p 2 (X 2 ) (M) r(b). (2.3) X 1 C n 1 m, X 2 C n 2 m (a) There exist X 1 C n1 m and X 2 C n2 m such that p 1 (X 1 ) p 2 (X 2 ) is nonsingular if and only if r(m) m. (b) p 1 (X 1 ) p 2 (X 2 ) is nonsingular for all X 1 C n1 m and X 2 C n2 m if and only if r( A 1 A 2 ) = m and B =. (c) There exist X 1 C n1 m and X 2 C n2 m such that p 1 (X 1 ) = p 2 (X 2 ) if and only if r(m) = 2r(B). (d) p 1 (X 1 ) = p 2 (X 2 ) for all X 1 C n1 m and X 2 C n2 m if and only if A 1 = A 2 and B =. (e) There exist X 1 C n1 m and X 2 C n2 m such that p 1 (X 1 ) > p 2 (X 2 ) (p 1 (X 1 ) < p 2 (X 2 )) if and only if i + (M) = m (i (M) = m). (f) p 1 (X 1 ) > p 2 (X 2 ) (p 1 (X 1 ) < p 2 (X 2 )) for all X 1 C n1 m and X 2 C n2 m if and only if i (M) = m (i + (M) = m). (g) There exist X 1 C n1 m and X 2 C n2 m such that p 1 (X 1 ) p 2 (X 2 ) (p 1 (X 1 ) p 2 (X 2 )) i (M) = r(b) (i + (M) = r(b)). (h) p 1 (X 1 ) p 2 (X 2 ) (p 1 (X 1 ) p 2 (X 2 )) for all X 1 C n1 m and X 2 C n2 m if and only if A 1 A 2 and B = (A 1 A 2 and B =.) 1

Proof The difference of p 1 (X 1 ) and p 2 (X 2 ) in (2.26) can be written as p 1 (X 1 ) p 2 (X 2 ) = A 1 A 2 + [ B 1, B 2 [ X1 X 2 + [ 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) (2.3). Results (a) (h) follow from (2.27) (2.3) and Lemma1.2. The following result was recently shown in [63. Lemma 2.6 Let A C m H, B Cm n and C C m p be given, and denote N = [ B, C. Then, max X C n H, Y Cp H max X C n H, Y Cp H [ i ± ( A BXB CY C A N ) N [ i ± ( A BXB CY C A N ) = r[ A, N i N Combining Theorem 2.2 with Lemma 2.6 leads to the following result., (2.32) Theorem 2.7 Let A C m H, B Cm n and C C m p and D C m q be given, and denote. (2.33) p(x, Y, Z ) = A BX (BX) CY C DZD, N = [ B, C, D. (2.34) Then, max X C n m, Y C p H, Z Cq H, (2.35) [ A N i ± [ p(x, Y, Z ) N [ A N A N B r(b) i N min i ± [ p(x, Y, Z ) = r X C n m, Y C p H, Z Cq H. (2.36) If A, then max i + [ p(x, Y, Z ) = r[ A, B, C, D, (2.37) X C n m, Y C p H, Z Cq H max i [ p(x, Y, Z ) = r[ B, C, D, (2.38) X C n m, Y C p H, Z Cq H min i ± [ p(x, Y, Z ) = r[ A, B, C, D r[ B, C, D, (2.39) X C n m, Y C p H, Z Cq H min X C n m, Y C p H, Z Cq H i ± [ p(x, Y, Z ) =. (2.4) Proof Applying (2.3) and (2.5) to (2.34) gives A CY C max i ±[ p(x, Y, Z ) = i DZD B ± X B, (2.41) A CY C min i ±[ p(x, Y, Z ) = i DZD B ± X B r(b). (2.42) Note that A CY C DZD B B = A B C B Y [ C D, Z[ D,. (2.43) Applying (2.32) and (2.33) to (2.43) gives ( ) A B C max i ± Y, Z B Y [ C D, Z[ D A N, = i ± N, ( ) [ A B C min i ± Y, Z B Y [ C D, Z[ D A N A N, = r B i N. 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. For instance, if R(B) R[ C, D, then max X C n m, Y C p H, Z Cq H, (2.44) [ A N i ± [ p(x, Y, Z ) N [ A N A N B r(b) i N min i ± [ p(x, Y, Z ) = r X C n m, Y C p H, Z Cq H, (2.45) where N = [ C, D. We shall use (2.44) and (2.45) in Section 4 to characterize the existence of nonnegative definite 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 for X C n m, where C C p n and D C p m are given. Also, denote Then, Hence, M = A B D B C D C, N = [ B C. max r[ p(x) = min{ m, r(m) 2r(C)}, (2.46) CX=D r[ p(x) = r(m) 2r(N), (2.47) min CX=D max ±[ p(x) (M) r(c), CX=D (2.48) min ±[ p(x) (M) r(n). CX=D (2.49) (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) > (< ) if and only if i + (M) = r(c)+m (i (M) = r(c) + m). (g) p(x) (< ) 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) ( ) if and only if i (M) = r(n) (i + (M) = r(n)). (i) Any solution of CX = D satisfying p(x) ( ) 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+F C V, where V C n m is arbitrary. Substituting it into p(x) gives rise to p(x) = A BC D (BC D) BF C V V (BF C ). (2.5) 12

Applying (2.1), (2.2), (2.3) and (2.5) to it gives max r[ p(x) = max CX=D min r[ p(x) = min CX=D max i ±[ p(x) = CX=D r[ A V C BC D (BC D) BF C V V (BF C ) n m { [ A BC = min m, r D (BC D) BF C (BF C ) min i ±[ p(x) = min CX=D r[ A BC D (BC D) BF C V V (BF C ) V C [ n m A BC = r D (BC D) BF C (BF C ) }, (2.51) 2r(BF C ), (2.52) max i ±[ A BC D (BC D) BF C V V (BF C ) V C n m [ A BC D (BC D) BF C (BF C ), (2.53) V i ±[ A BC D (BC D) BF C V V (BF C ) [ A BC D (BC D) BF C (BF C ) r(bf C ). (2.54) Applying (1.14) and (1.15), and simplifying by CC D = D and EBCMOs, we obtain A BC i D (BC D) BF C ± (BF C ) = i ± A BC D (BC D) B B C r(c) C A B D B C r(c) D C (M) r(c), (2.55) A BC r D (BC D) BF C (BF C ) = r(m) 2r(C), (2.56) B r(bf C ) = r r(c) = r(n) r(c). (2.57) C 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 solutions to some matrix equations As some applications of results in Section 2, we derive in this section the extremal values of the ranks and partial inertias of for the Hermitian parts of solutions of the two equations in (1.18) and (1.2), and give some direct consequences of these extremal values. Theorem 3.1 Let A, B C m n be given, and assume the matrix equation AX = B is solvable for X C n n. Then, (a) The maximal value of the rank of X + X is max r( X + AX=B X ) = min{ n, 2n + r( AB + BA ) 2r(A)}. (3.1) (b) The minimal value of the rank of X + X is A matrix X C n n satisfying (3.2) is given by where U = U C n n is arbitrary. min r( X + AX=B X ) = r( AB + BA ). (3.2) X = A B (A B) + A AB (A ) + F A UF A, (3.3) 13

(c) The maximal values of the partial inertia of X + X are max i ±( X + X ) = n + i ± ( AB + BA ) r(a). (3.4) AX=B A matrix X C n n satisfying the two formulas in (3.4) is given by where U = U C n n is arbitrary. X = A B (A B) + A AB (A ) ± F A + F A UF A, (3.5) (d) The minimal values of the partial inertia of X + X are min i ±( X + X ) ( AB + BA ). (3.6) AX=B A matrix X C n 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 =, i.e., AX = B has a skew-hermitian solution, if and only if AB + BA =. Such a solution is given by where U = U C n n is arbitrary. X = A B (A B) + A AB (A ) + F A UF A, (3.7) (h) Any solution of AX = B satisfying X + X = 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 >, i.e., AX = B has a Re-positive definite solution, if and only if i + ( AB + BA ) = r(a). Such a solution is given by where U = U C n n is arbitrary. X = A B (A B) + A AB (A ) + F A + F A UF A, (3.8) (k) AX = B has a solution X C n n satisfying X + X <, i.e., AX = B has a Re-negative definite solution, if and only if i ( AB + BA ) = r(a). Such a matrix is given by where U = U C n n is arbitrary. X = A B (A B) + A AB (A ) F A + F A UF A, (3.9) (l) Any solution of AX = B satisfying X + X (i ( AB + BA ) = n). > (< ) if and only if i + ( AB + BA ) = n (m) AX = B has a solution satisfying X +X, i.e., AX = B has a Re-nonnegative definite solution, if and only if AB + BA. Such a matrix is given by X = A B (A B) + A AB (A ) + F A (U + W )F A, (3.1) where U = U C n n and W C n n are arbitrary. (n) AX = B has a solution X satisfying X + X, i.e., AX = B has a Re-non-positive definite solution, if and only if AB + BA. Such a matrix is given by X = A B (A B) + A AB (A ) + F A (U W )F A, (3.11) where U = U C n n and W C n n are arbitrary. (o) Any solution of AX = B satisfying X + X ( ) if and only if AB + BA and r(a) = n (AB + BA 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 = and B = I m, and replacing C and D with A and B in (2.46) (2.49), we obtain (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 + F A UF A F A UF A Also, note that = A ( AB + BA )(A ). (3.12) 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 easy to verify that (3.5) satisfies AX = B. Substituting (3.5) into X + X gives X + X = A ( AB + BA )(A ) ± 2F A. (3.14) Also, note that R(A ) R(F A ) = {}. Hence, (3.14) implies that i ± ( X + X ) ( AB + BA ) + i ± (±F A ) = 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 definite solutions, we are also able to derive from (2.46) (2.49) the solutions of AX = B that satisfies X +X > P (< P, P, P ). In what follows, we derive the extremal values of the ranks and partial inertias of the Hermitian parts of solutions of the matrix equation AXB = C. Theorem 3.2 Let A C m n, B C n p and C C m p be given, and assume that the matrix equation AXB = C is solvable for X C n n. Also, denote C A M = C B, N = [ A, B. A B Then, Hence, max AXB=C X ) = min {n, 2n + r(m) 2r(A) 2r(B)}, (3.15) min AXB=C X ) = r(m) 2r(N), (3.16) max ±(X + X ) = n + i (M) r(a) r(b), AXB=C (3.17) min ±(X + X ) = i (M) r(n). AXB=C (3.18) (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 C n n satisfying X + X = i.e., AXB = C has a skew-hermitian solution, 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 r(a) = r(b) = n. subject to AXB = C is invariant if and only if r(m) = 2r(N) + n or 15

(f) AXB = C has a solution satisfying X + X > (X + X < ), i.e., AXB = C has a Repositive 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 > (X + X < ) if and only if i (M) = r(n) + n (i + (M) = r(n) + n). (h) AXB = C has a solution X C n n satisfying X + X (X + X ), i.e., AXB = C has a Re-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 (X +X ) 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 the solution of AXB = C can be written as X + X = A CB + (A CB ) + [ F A, E B V + V [ F A, E B, (3.19) where V = V1 V2 C 2n n is arbitrary. Applying (2.1), (2.2), (2.3) and (2.5) to (3.22) gives max r(x + AXB=C X ) = max r ( A CB + (A CB ) + [ F A, E B V + V [ F A, E B ) V = min {n, r(j)}, (3.2) min r(x + AXB=C X ) = min r( A CB + (A CB ) + [ F A, E B V + V [ F A, E B ) V = r (J) 2r[ F A, E B, (3.21) max i ±(X + X ( ) = max i ± A CB + (A CB ) + [ F A, E B V + V [ F A, E B ) AXB=C V (J), (3.22) min i ±(X + X ) = min i ( ± A CB + (A CB ) + [ F A, E B V + V [ F A, E B ) AXB=C V (J) r[ F A, E B, (3.23) where J = A CB + (A CB ) F A E B F A E B. 16

Applying (1.14) and simplifying by AA CB B = AA C = CB B = C and EBCMOs, we obtain i ± (J) A CB + (A CB ) F A E B F A E B A CB + (A CB ) I n I n I n A I n B A B I n I n 1 2 (CB ) 1 2 A C I n A I n B 1 2 CB A 1 2 (A C) B I n I n A B A 1 2 CB A + 1 2 A(B ) C 1 2 C B 1 2 C = n + i ± A B A C r(a) r(b) B C = n + i Applying (1.5) and simplifying by EBMOs, we obtain r(a) r(b) r(a) r(b) r(a) r(b) C A C B r(a) r(b), (3.24) A B r[ F A, E B = r I n I n A r(a) r(b) = r I n A r(a) r(b) B B Adding the two equalities in (3.24) gives i ± = n + r[ A, B r(a) r(b). (3.25) A CB + (A CB ) F A E B C A F A = 2n + r C B 2r(A) 2r(B), (3.26) E B A B Substituting (3.24), (3.25) and (3.26) into (3.2) (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 and generalized inverse of matrices among them. In comparison, Theorem 3.2 shows that the existence of skew-hermitian solution and Re-definite solution of the matrix equation AXB = C can be characterized by some explicit equalities for the rank and partial inertia of a Hermitian block matrix composed by the given matrices in the equation. Theorem 3.3 Let A C m n, B C n p, C C m p and P C n H be given, and assume that the matrix equation AXB = C is solvable for X C n n. Also, denote M = AP A C A C B, N = [ A, B. A B 17

Then, max AXB=C X P ) = min {n, 2n + r(m) 2r(A) 2r(B)}, (3.27) min AXB=C X P ) = r(m) 2r(N), (3.28) max ±( X + X P ) = n + i (M) r(a) r(b), AXB=C (3.29) min ±( X + X P ) = i (M) r(n). AXB=C (3.3) 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 C n 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 or r(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 C n n satisfying X + X P (X + X P ) if and only if i (M) = r(n) (i + (M) = r(n)). (i) All solutions of AXB = C satisfy X +X P (X +X 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 can be written as X + X P = A CB + (A CB ) P + [ F A, E B V + V [ F A, E B, (3.31) where V C 2n n ia arbitrary. Applying (2.1), (2.2), (2.3) and (2.5) to (3.31) gives max r( X + AXB=C X P ) = max r ( A CB + (A CB ) P + [ F A, E B V + V [ F A, E B ) V = min {n, r(j)}, (3.32) min r( X + AXB=C X P ) = min r( A CB + (A CB ) P + [ F A, E B V + V [ F A, E B ) V = r(j) 2r[ F A, E B, (3.33) max i ±( X + X ( P ) = max i ± A CB + (A CB ) P + [ F A, E B V + V [ F A, E B ) AXB=C V r(j), (3.34) min i ±( X + X P ) = min i ( ± A CB + (A CB ) P + +[ F A, E B V + V [ F A, E B ) AXB=C V (J) r[ F A, E B, (3.35) where J = A CB + (A CB ) P F A E B F A E B. 18

Applying (1.14) and simplifying by AA CB B = AA C = CB B = C and EBCMOs, we obtain i ± (J) A CB + (A CB ) P F A E B F A E B A CB + (A CB ) P I n I n I n A I n B A B r(a) r(b) I n I n 1 2 (CB ) + 1 4 P A 1 2 A C + 1 4 P B I n A I n B 1 2 CB + 1 4 AP A 1 2 (A C) + 1 4 B P B r(a) r(b) I n I n A B A 1 2 CB A + 1 2 A(B ) C 1 2 AP A 1 2 C 1 4 AP B B 1 2 C 1 4 B P A r(a) r(b) A B = n + i ± A AP A C r(a) r(b) B C AP A C A = n + i C B r(a) r(b) A 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) (3.3). Results (a) (j) follow from (3.27) (3.3) 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 C m m. Then, Hence, min r[ A + (A ) = r( A + A ) + 2r(A) 2r[ A, A, (3.38) A min i ±[ A + (A ) ( A + A ) + r(a) r[ A, A. (3.39) A (a) There exists an A such that A + (A ) = if and only if r( A + A ) + 2r(A) = 2r[ A, A. (b) There exists an A such that A + (A ) if and only if i + ( A + A ) + r(a) = r[ A, A. (c) There exists an A such that A + (A ) if and only if i ( A + A ) + r(a) = r[ A, A. 19

4 Extremal values of ranks/inertias of Hermitian solutions to some matrix equations Hermitian solutions and definite solutions of the matrix equations AX = B and AXB = C were considered in the literature, and various results were derived; see, e.g., [3, 33. In this section, we derive some new results on Hermitian and definite solutions of AXB = C through the matrix rank/inertia methods. Theorem 4.1 Let A C m n, B C n p and C C m p be given, and assume that the matrix equation AXB = C is solvable for X C n n. Then, Hence, the following statements are equivalent: min r( X AXB=C X ) = r C A C B 2r[ A, B. (4.1) B A (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) C A R(C) R(A), R(C ) R(B ), r C B = 2r[ A, B. (4.3) B A In this case, the general Hermitian solution to AXB = C can be written as X = 1 2 (Y + Y ), (4.4) where Y is the common solution to (4.2), or equivalently, X = 1 2 (Y + Y ) + E G U 1 + (E G U 1 ) + F A U 2 F A + E B U 3 E B, (4.5) where Y is a special common solution to (4.2), G = [ A, B, and three matrices U 1 C n n, U 2, U 3 C n H are arbitrary. Proof Note from (1.21) that the difference X X for the general solution of AXB = C can be written as where V = X X = A CB (A CB ) + F A V 1 + V 2 E B (F A V 1 ) (V 2 E B ) = A CB (A CB ) + [ F A, E B V + V [ F A, E B, V1 V2 is arbitrary. Applying (1.5) and simplifying by EMBOs, we obtain r[ F A, E B = r I n I n A r(a) r(b) B = r I n A r(a) r(b) = n + r[ A, B r(a) r(b). (4.6) B 2

Applying (2.2) to it and simplifying by (1.15), (4.6), AA C = CB B = C and EMBOs, we obtain min r( X AXB=C X ) = min r( A CB (A CB ) + [ F A, E B V + V [ F A, E B ) V = r A CB (A CB ) F A E B F A 2r[ F A, E B E B A CB (A CB ) I n I n I n A = r I n B A B I n I n A = r I n B CB A 2n 2r[ A, B (A C) B I n I n A B = r A C 2n 2r[ A, B B C = r C A C B 2r[ A, B, B A 2n 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 X, then it satisfies A X B = C, that is to say, the pair of equations in (4.2) have a common solution X. 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 = 1 2 (AY B + AY B) = 1 (C + C) = C. 2 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 X to AXB = C is a common solution to (4.2), and can be written as X = 1 2 (X + X ). 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 = Y + E G V 1 + V 2 E G + F A V 3 F A + E B V 4 E B, where Y is a special solution of the pair, G = [ A, B, and V 1,..., V 4 C n n are arbitrary. Substituting it into (4.4) yields X = 1 2 (Y + Y ) = 1 2 (Y + Y ) + 1 2 E G(V 1 + V 2 ) + 1 2 (V 1 + V 2 )E G + 1 2 F A(V 3 + V 3 )F A + 1 2 E B(V 4 + V 4 )E B = 1 2 (Y + Y ) + E G U 1 + (E G U 1 ) + F A U 2 F A + E B U 3 E B, where U 1 C n n, U 2, U 3 C n H are arbitrary, establishing (4.5). Theorem 4.2 Let A C m n, B C n p, C C m p and P C n H be given, and assume that the matrix equation AXB = C has a solution X C n H. Also, denote M 1 = AP A C A C B P B B, M 2 = C A C B, A B A B 21

A AP A C N 1 = B C B, N P B 2 = A C B C. Then, max i ± ( X P ) = i (M 1 ) + n r(a) r(b), (4.7) AXB=C, X C n H min AXB=C, X C n H i ± ( X P ) = r(n 1 ) i ± (M 1 ), (4.8) max i ± (X) = i (M 2 ) + n r(a) r(b), (4.9) AXB=C, X C n H min AXB=C, X C n H i ± (X) = r(n 2 ) i ± (M 2 ). (4.1) Hence, (a) AXB = C has a solution X > P (X < P ) if and only if i (M 1 ) = r(a) + r(b) (i + (M 1 ) = r(a) + r(b)). (b) AXB = C has a solution X P (X P ) if and only if i (M 1 ) = r(n 1 ) (i + (M 1 ) = r(n 1 )). (c) AXB = C has a solution X > (X < ) if and only if i (M 2 ) = r(a) + r(b) (i + (M 2 ) = r(a) + r(b)). (d) All solutions of AXB = C satisfy X > (X < ) if and only if i + (M 2 ) = r(n 2 ) n (i (M 2 ) = r(n 2 ) n). (e) AXB = C has a solution X (X ) if and only if i (M 2 ) = r(n 2 ) (i + (M 2 ) = r(n 2 )). (f) All solutions of AXB = C satisfy X (X ) if and only if i + (M 2 ) = r(a) + r(b) n (i (M 2 ) = r(a) + r(b) n). Proof We first show that the set inclusion R(F G ) R[ F A, E B. Applying (1.5) to [ E G, F A, E B and [ F A, E B and simplifying by EBMOs, we obtain I n I n I n r[ E G, F A, E B = r G A r(g) r(a) r(b) B I n = r G G A r(g) r(a) r(b) B A = r B A + n r(g) r(a) r(b) = n + r(g) r(a) r(b). B 22

Combining it with (4.6) leads to r[ F G, F A, E B = r[ F A, E B, i.e., R(F G ) R[ F A, E B. In this case, applying (2.44) and (2.45) to (4.5) gives max i ± ( X P ) = max i ± [ X P + E G U 1 + (E G U 1 ) + F A U 2 F A + E B U 3 E B AXB=C, X C n H U 1, U 2, U 3 X P F A E B F A, (4.11) E B min i ± ( X P ) = min i ± [X P + E G U 1 + (E G U 1 ) + F A U 2 F A + E B U 3 E B AXB=C,X C n H U 1, U 2, U 3 X P F = r A E B i E G X P F A E B F A r(e G ). (4.12) E B Applying (1.14) and simplifying by AX B = C and EBCMOs, we obtain i ± X P F A E B F A E B X P I n I n I n A I n B A B r(a) r(b) I n I n 1 4 X A + 1 4 P A 1 4 X B + 1 4 P B I n A I n B 1 4 AX + 1 4 AP A 1 4 B X + 1 4 B P B r(a) r(b) I n I n A B A 1 2 AX A 1 2 AP A 1 4 C 1 4 AP B r(a) r(b) B 1 4 C 1 4 B P A A B A 1 2 AP A 1 2 C + n r(a) r(b) B 1 2 C 1 2 B P B A B A AP A C + n r(a) r(b) B C B P B = i (M 1 ) + n r(a) r(b), 23

and applying (1.15) and simplifying by AX B = C and EBMOs, we obtain X P I n I n X P F r A E B = r I n G E G A r(g) r(a) r(b) B I n I n P G X G = r I n A r(g) r(a) r(b) B = r I n A AX G AP G + n r(g) r(a) r(b) B A AX A = r AP A C AP B B + 2n r(g) r(a) r(b) A AP A C AP B = r B C + 2n r(g) r(a) r(b) A AP A C = r B C B + 2n r(g) r(a) r(b) P B = r(n 1 ) + 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 = in (4.7) and (4.8) yields (4.9) and (4.1), respectively. Results (a) (f) follow from (4.7) (4.1) and Lemma 1.4. A special case of the matrix equation AXB = C is the matrix equation AXA = C, which was studied by many authors; see, e.g., [1, 18, 2, 33, 38, 41. Applying Theorems 4.1 and 4.2 to AXA = C leads to the following result. Corollary 4.3 Let A C m n, C C m H and P Cn H be given. Then, (a) The matrix equation AXA = C (4.13) has a solution X C n H if and only if R(C) R(A). In this case, the general Hermitian solution to (4.13) can be written as X = A C(A ) + F A U + (F A U), (4.14) where U C n n is arbitrary. (b) Under R(C) R(A), max i ± ( X P ) = n + i ± ( C AP A ) r(a), (4.15) AXA =C, X C n H min AXA =C, X C n H i ± ( X P ) ( C AP A ), (4.16) max i ± (X) = n + i ± (C) r(a), (4.17) AXA =C, X C n H min AXA =C, X C n H i ± (X) (C). (4.18) (c) Under R(C) R(A), (4.13) has a solution X > P (X < P ) if and only if i + ( C AP A ) = r(a) (i ( C AP A ) = r(a)). (d) Under R(C) R(A), (4.13) has a solution X P (X P ) if and only if C AP A (C AP A ). (e) Under R(C) R(A), (4.13) has a solution X > (X < ) if and only if C and r(a) = r(c) (C and r(a) = r(c)). (f) Under R(C) R(A), (4.13) has a solution X (X ) if and only if C (C ). 24

Corollary 4.4 Let A C m m be given. Then, (a) A has a Hermitian g-inverse if and only if In this case, r( A A ) = 2r[ A, A 2r(A). (4.19) max r(a ) = min{m, r( A + A ) + 2m 2r(A)}, (4.2) A min r(a ) = 2r(A) r( A + A ), (4.21) A max i ±(A ) ( A + A ) + m r(a), (4.22) A min i ±(A ) = r(a) i ( A + A ). (4.23) A (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 > there exists an A i + ( A + A ) = r(a). (e) There exists an A < there exists an A i ( A + A ) = r(a). As is well known, one of the fundamental concepts in matrix theory is the partition of a matrix. Many algebraic properties of a matrix and is operations can be derived from the submatrices in its partitions and their operations. In order to reveal more properties of Hermitian solutions to (4.13), we partition the unknown Hermitian matrix X in (4.13) into a 2 2 block form Consequently, (4.13) can be rewritten as X = X1 X 2 X2. X 3 X1 X [ A 1, A 2 2 A 1 X2 X 3 A = C, (4.24) 2 where A 1 C m n1, A 2 C m n2, X 1 C n1 H, X 2 C n1 n2 and X 3 C n2 H with n 1 + n 2 = n. In what follows, we derive the extremal values of the ranks and partial inertias of the submatrices in a Hermitian solution to (4.24). Note that X 1, X 2, X 3 can be rewritten as X 1 = P 1 XP 1, X 2 = P 1 XP 2, X 3 = P 2 XP 2, (4.25) where P 1 = [ I n1, and P 2 = [, I n2. Substituting the general solution in (4.14) into (4.25) yields X 1 = P 1 X P1 + P 1 F A V 1 + V1 F A P1, (4.26) X 3 = P 2 X P2 + P 2 F A V 2 + V2 E B P2, (4.27) where X = A C(A ), V = [ V 1, V 2. For convenience, we adopt the following notation for the collections of the submatrices X 1 and X 3 in (4.24): { S 1 = S 3 = X j C n1 H { X 3 C n2 H [ A X1 X 1, A 2 2 A 1 X2 X 3 A 2 [ A X1 X 1, A 2 2 A 1 X2 X 3 A = C 2 } = C Theorem 4.5 Suppose that the matrix equation (4.24) is consistent. Then,, (4.28) }. (4.29) 25