Yongge Tian. China Economics and Management Academy, Central University of Finance and Economics, Beijing , China

On global optimizations of the rank and inertia of the matrix function A 1 B 1 XB 1 subject to a pair of matrix equations B 2 XB 2, B XB = A 2, A Yongge Tian China Economics and Management Academy, Central University of Finance and Economics, Beijing 181, China Abstract. For a given linear matrix function A 1 B 1XB 1, where X is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations B 2XB 2 = A 2 and B XB = A. As applications, we give necessary and sufficient conditions for the triple matrix equations B 1XB 1 = A 1, B 2XB 2 = A 2 and B XB = A to have a common Hermitian solution. In addition, we discuss the global optimizations on the rank and inertia of the common Hermitian solution of the pair of matrix equations B 2XB 2 = A 2 and B XB = A. AMS subject classifications: 15A, 15A9, 15A24; 65K1; 65K15 Key words: Matrix function; matrix equation; common Hermitian solution; rank; inertia; maximization; minimization; Löwner partial ordering 1 Introduction 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 definite (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 definite (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. 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. 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. Both i + (A) and i (A), usually called the partial inertia, can easily be computed by elementary congruence matrix operations. For a Hermitian matrix A, the equality r(a) = i + (A) + i (A) holds. In recent years, some optimization problems on the global maximum and minimum ranks and inertias of the following linear matrix functions (LMFs) A BXB, A BX (BX), A BXC (BXC), (1.1) and their applications were studied, where A is a given complex Hermitian matrix, B and C are given complex matrices, and X is a variable matrix of appropriate size; see 14, 15, 16, 17, 18, 28, 1, 2,. As a generalization, we address in this paper the following optimization problem: Problem 1.1 For a given LMF A 1 B 1 XB 1 and a pair of matrix equations B 2 XB 2, B XB = A 2, A that have a a common Hermitian solution, give formulas for calculating the following global E-mail Address: yongge.tian@gmail.com

maximum and minimum ranks and inertias max X C n H min X C n H r( A 1 B 1 XB 1 ) s.t. B 2 XB 2, B XB = A 2, A, (1.2) r( A 1 B 1 XB 1 ) s.t. B 2 XB 2, B XB = A 2, A, (1.) max i ± ( A 1 B 1 X ) s.t. B 2 XB2, B XB = A 2, A, (1.4) X C n H min i ± ( A 1 B 1 X ) s.t. B 2 XB2, B XB = A 2, A. (1.5) X C n H We shall use some pure algebraic operations on matrices to derive a group of analytical formulas for calculating the global maximum and minimum values of the objective functions in (1.2) (1.5), and then to present a variety of valuable consequences of these formulas. In particular, we shall use the minimum rank formula for (1.) to derive necessary and sufficient conditions for the triple matrix equations B 1 XB 1 = A 1, B 2 XB 2 = A 2, B XB = A (1.6) to have a common Hermitian solution. The rank and inertia of a Hermitian matrix are two basic concepts in matrix theory for describing the dimension of the row/column vector space and the sign distribution of the eigenvalues of the matrix, which are well understood and are easy to compute by the well-known elementary or congruent matrix operations. These two quantities play an essential role in characterizing algebraic properties of Hermitian matrices. Because the rank and inertia of a matrix are finite nonnegative integers, the global maximum and minimum values of the rank and inertia of a matrix expression always exist no matter what the domains of variable entries in the matrix expression are given. The extremal ranks and inertias of a matrix expression can directly be used to characterize some fundamental algebraic properties of the matrix expression, for example, (I) the maximum and minimum dimensions of the row and column spaces of the matrix expression; nonsingularity of the matrix expression when it is square; (III) solvability of the corresponding matrix equation; (IV) rank, inertia and range invariance of the matrix expression; (V) definiteness of the matrix expression when it is Hermitian; etc. Since variable entries in a matrix function are often regarded as continuous variables in some constrained sets, while the objective functions the rank and inertia of the matrix function take values only from a finite set of nonnegative integers, Hence, (1.2) (1.5) can be regarded as continuous-integer optimization problems subject to equality constraints. This kind of nonsmooth optimization problems cannot be solved by using various optimization methods for solving continuous or discrete cases. There is no rigorous mathematical theory for solving a general rank and inertia optimization problem due to the discontinuity and nonconvexity of rank and inertia of matrix. In fact, it has been realized that rank and inertia optimization problems have deep connections with computational complexity, are regarded as NP-hard in general settings; see, e.g., 1, 2,, 4, 6, 7, 8, 11, 2, 2, 25. The following are some known results for ranks and inertias of matrices and their usefulness, which will be used in the latter part of this paper. Lemma 1.2 (28) Let H be a set consisting of Hermitian matrices over C m H. Then, (a) H has a matrix X > (X < ) if and only if max X H i +(X) = m (b) All X H satisfy X > (X < ) if and only if min X H i +(X) = m (c) H has a matrix X (X ) if and only if min X H i (X) = (d) 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 2

Lemma 1. (19) Let A C m n, B C m p and C C q n. Then, the following rank expansion formulas hold r A, B = r(a) + r(e A B) = r(b) + r(e B A), (1.7) A r = r(a) + r(cf C A ) = r(c) + r(af C ), (1.8) A B r = r(b) + r(c) + r(e C B AF C ). (1.9) Three useful rank expansion formulas derived from (1.9) are A B A B r = r(p ) + r C P E P C A B r C A BFQ = r(q) + r C Q A B r C P Q = r(p ) + r(q) + r A BFQ E P C, (1.1), (1.11) We shall use them in Section 2 to simplify ranks of block matrices involving E P and F Q. Lemma 1.4 (28) Let A C m H, B Cm n, D C m n, and let A B A B U = B, V = B. D Then, the inertias of U and V can be expanded as (a) If A, then (b) If A, then (c) If R(B) R(A), then. (1.12) i ± (U) = r(b) + i ± (E B AE B ), (1.1) E i ± (V ) = i ± (A) + i A B ± B E A D B A. (1.14) B i + (U) = r A, B, i (U) = r(b), r(u) = r A, B + r(b). (1.15) i + (U) = r(b), i (U) = r A, B, r(u) = r A, B + r(b). (1.16) i ± (V ) = i ± (A) + i ± ( D B A B ), r(v ) = r(a) + r( D B A B ). (1.17) (d) If R(B) R(A) = {} and R(B ) R(D) = {}, then i ± (V ) = i ± (A) + i ± (D) + r(b), r(v ) = r(a) + 2r(B) + r(d). (1.18) Three expansion formulas derived from (1.1) are A B A B A BF i P ± F P B = i ± B P A BF r(p ), r P F P P B = r B P 2r(P ). (1.19) P We shall use them to simplify the inertias of block Hermitian matrices that involve F P = I P P. Lemma 1.5 Let A C m n and B C m H be given. Then,

(a) 5, 9 The matrix equation AXA = B has a solution X C n H if and only if R(B) R(A), or equivalently, AA B = B. (b) 28 Under AA B = B, the general Hermitian solution of AXA = B can be written in the following two forms where U C n H and V Cn n are arbitrary. X = A B(A ) + U A AUA A, (1.2) X = A B(A ) + F A V + V F A, (1.21) More results on properties of solutions of AXA = B can be found in 15, 18. Lemma 1.6 Let A j C mj n, B j C p qj and C j C mj qj be given, j = 1, 2. Then, (a) 24 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. (1.2) B 1 B 2 (b) 27 Under (1.27), the general common solution to (1.26) can be written in the following parametric form X = X + F A V 1 + V 2 E B + F A1 V E B2 + F A2 V 4 E, (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 Lemma 1.7 (17) Let A C m H, B Cm p and C C q m be given, and let A B A C M 1 = B, M 2 =, (1.25) C N = A, B, C A B C A B C, N 1 = B, N 2 =. (1.26) C Then, the global maximum and minimum rank and inertias of A BXC (BXC) are given by where max r A BXC (BXC) = min {r(n), r(n 1 ), r(n 2 )}, (1.27) X C p q min r A BXC (BXC) = 2r(N) + max{ s 1, s 2, s, s 4 }, (1.28) X C p q max i ± A BXC (BXC) = min{i ± (M 1 ), i ± (M 2 )}, (1.29) X C p q min i ± A BXC (BXC) = r(n) + max{ i ± (M 1 ) r(n 1 ), i ± (M 2 ) r(n 2 ) }, (1.) X C p q s 1 = r(m 1 ) 2r(N 1 ), s 2 = r(m 2 ) 2r(N 2 ), s = i + (M 1 ) + i (M 2 ) r(n 1 ) r(n 2 ), s 4 = i + (M 1 ) + i (M 2 ) r(n 1 ) r(n 2 ). In particular, if R(C ) R(B), then { } A C max r A BXC X C (BXC) = min r A, B, r, (1.1) p q C A C min r A BXC A B (BXC) = 2r A, B + r 2r, (1.2) X C p q C C max i ± A BXC (BXC) A C = i ±, (1.) X C p q C min i ± A BXC (BXC) A C A B = r A, B + i ± r. (1.4) X C p q C C 4

The matrices Xs that satisfy (1.27) (1.) (namely, the global maximizers and minimizers of the objective rank and inertia functions) are not necessarily unique and their expressions were also given in 17 by using certain simultaneous decomposition of the three given matrices. Observe that the righthand sides of (1.27) (1.) are represented in analytical forms of the ranks and inertias of the five given block matrices, we can easily use them to derive extremal ranks and inertias of some general linear and nonlinear matrix functions. In these cases, combining the rank and inertia formulas obtained with the assertions in Lemma 1.1 may yield various conclusions on algebraic properties of linear and nonlinear matrix functions. 2 The global maximum and minimum ranks and inertias of A 1 B 1 XB 1 subject to a consistent matrix equation Because of the noncommutativity of matrix multiplications, solving matrix equations has been a challenging topic of study in linear algebra. If some matrix equations of appropriate sizes are given together, it is natural to ask whether the equations have a possible common solution. For example, if two Hermitian matrix equations B 1 X 1 = A 1 and B 2 X 2 B2 = A 2, are given, where X 1 and X 2 have the same size, and each of them has a solution. In such a case, it would be of interest to seek relations between the Hermitian solutions of the two equations. As mentioned in the previous section, the existence of solution of a matrix equation can be characterized by the minimum rank of the corresponding matrix expression. Further, relations between two matrix equations can also be characterized by rank/methods. Some recent work on the rank and inertia method in the investigation of Hermitian matrix equations can be found in 14, 15, 16, 17, 28,. In 17, the extremal ranks and inertias of A 1 B 1 X subject to a consistent matrix equation B 2 XB2 = A 2 were studied and the following results were obtained. Lemma 2.1 (17) Let A i C mi H and B i C mi n be given for i = 1, 2, and suppose that the matrix equation B 2 XB2 = A 2 has a Hermitian solution. Also, let A 1 B 1 S 2 = { X C n H B 2 XB2 = A 2 }, M = A 2 B 2. (2.1) B2 Then, the global maximum and minimum rank and inertias of A 1 B 1 XB 1 s.t. X S are given by Hence, max r( A 1 B 1 X ) = min {r A 1, B 1, r(m) 2r(B 2 )}, (2.2) 2 min r( A 1 B 1 X A1 B ) = 2r A 1, B 1 2r 1 2 B2 + r(m), (2.) max i ± ( A 1 B 1 X ) = i ± (M) r(b 2 ), (2.4) 2 min i ± ( A 1 B 1 X A1 B ) = r A 1, B 1 r 1 2 B2 + i ± (M). (2.5) (a) There exists an X C n H such that B 2XB 2 = A 2 and A 1 B 1 XB 1 is nonsingular if and only if r A 1, B 1 = m 1 or r(m) = 2r(B 2 ) + m 1. (b) There exists an X C n H such that B 1X = A 1 and B 2 XB2 = A 2 if and only if R(A 1 ) R(B 1 ) and R(A 2 ) R(B 2 ) and r(m) = 2r. (2.6) B 2 (c) There exists an X C n H such that B 2XB 2 = A 2 and A 1 B 1 XB 1 > if and only if i + (M) = r(b 2 ) + m 1. (d) There exists an X C n H such that B 2XB 2 = A 2 and A 1 B 1 XB 1 < if and only if i (M) = r(b 2 ) + m 1. 5

(e) There exists an X C n H such that B 2XB2 = A 2 and A 1 B 1 X if and only if A1 B R(A 2 ) R(B 2 ) and i + (M) = r 1 B2 r A 1, B 1. (2.7) (f) There exists an X C n H such that B 2XB2 = A 2 and A 1 B 1 X if and only if A1 B R(A 2 ) R(B 2 ) and i (M) = r 1 B2 r A 1, B 1. (2.8) Corollary 2.2 Let A i C mi H and B i C mi n be given with B i for i = 1, 2, and suppose that each of the two matrix equations B 1 X = A 1 and B 2 XB2 = A 2 has a Hermitian solution. Also denote the sets of all solutions of B 1 X = A 1 and B 2 XB2 = A 2 by S 1 and S 2 respectively. Then, the global maximum and minimum ranks and inertias of A 1 B 1 X subject to X S 2 are given by where M is as given in (2.1). In particular, max r( A 1 B 1 X ) = min {r(b 1 ), r(m) 2r(B 2 )}, (2.9) 2 min r( A 1 B 1 X ) = r(m) 2r, (2.1) 2 B 2 max i ± ( A 1 B 1 X ) = i ± (M) r(b 2 ), (2.11) 2 min i ± ( A 1 B 1 X ) = i ± (M) r, (2.12) 2 B 2 (a) S 2 S 1 if and only if r(m) = 2r(B 2 ). (b) S 2 = S 1 if and only if r(m) = 2r(B 1 ) = 2r(B 2 ). The above results can be used to derive algebraic properties of the submatrices in a solution to the matrix equation BXB = A. Rewrite BXB = A as X1 X B 1, B 2 2 B 1 X2 X B2 = A, (2.1) where B 1 C m n1, B 2 C m n2, X 1 C n1 H, X 2 C n1 n2 and X C n2 H with n 1 + n 2 = n. We next derive the global maximum and minimum ranks and inertias of the submatrices X 1 and X in a Hermitian solution to (2.1). Note that X 1, X 2, X in (2.1) can be rewritten as X 1 = P 1 XP 1, X 2 = P 1 XP 2, X = P 2 XP 2, (2.14) where P 1 = I n1, and P 2 =, I n2. For convenience, we adopt the following notation for the collections of the submatrices X 1 and X in (2.1): { } T 1 = X 1 C n1 H B X1 X 1, B 2 2 B 1 X2 X B2 = A = {X 1 = P 1 XP1 BXB = A}, (2.15) { } T = X C n2 H B X1 X 1, B 2 2 B 1 X2 X B2 = A = {X = P 2 XP2 BXB = A}. (2.16) Applying Corollary 2.2 to (2.15) and (2.16) gives the following results. Theorem 2. Suppose that the matrix equation (2.1) is consistent, and let T 1 and T be of the forms (2.15) and (2.16). Then, { } A B2 max r(x 1 ) = min n 1, r X 1 T 1 B2 2r(B) + 2n 1, (2.17) A B2 min r(x 1 ) = r X 1 T 1 B2 2r(B 2 ), (2.18) max i ± (X 1 ) = i ± X 1 T 1 A r(a) + n 1, (2.19) min i ± (X 1 ) = i ± X 1 T 1 A r(b 2 ), (2.2) 6

and { } A max r(x ) = min n 2, r X T 2r(B) + 2n 2, (2.21) A min r(x ) = r X T 2r(B 1 ), (2.22) C max i ± (X ) = i ± X T A r(a) + n 1 2, (2.2) C min i ± (X ) = i ± X T A r(b 1 1 ). (2.24) Hence, (c) Eq. (2.1) has a solution in which X 1 is nonsingular if and only if r A 2r(A) n 1. (d) The submatrix X 1 in any solution to (2.1) is nonsingular if and only if r A = 2r(B 2 )+n 1. (e) Eq. (2.1) has a solution in which X 1 = if and only if r A = 2r(B 2 ). (f) The submatrix X 1 in any solution to (2.1) satisfies X 1 = if and only if r A = 2r(A) 2n 1. (g) Eq. (2.1) has a solution in which X 1 > (X 1 < ) if and only if ( ) i + A = r(a) i A = r(a). (h) The submatrix X 1 in any solution to (2.1) satisfies X 1 > (X 1 < ) if and only if ( ) i + A = n 1 + r(b 2 ) i A = n 1 + r(b 2 ). (i) Eq. (2.1) has a solution satisfying X 1 (X 1 ) if and only if ( ) i A = r(b 2 ) i + A = r(b 2 ). (j) The submatrix X 1 in any solution to (2.1) satisfies X 1 (X 1 ) if and only if ( ) i A = r(a) n 1 i A = r(a) n 1. (k) The positive signature of X 1 in (2.1) is invariant the negative signature of X 1 (2.1) is invariant R(B 1 ) R(B 2 ) = {} and r(b 1 ) = n 1. The definiteness of the Hermitian solutions of a given consistent matrix equation is attractive topic in matrix theory and applications; see, e.g., 1, 9,, 5. From Corollary 2.2, we now can derive the existences of Hermitian solutions of AXA = B satisfying some inequalities. Corollary 2.4 (1) Let A C m n, B C m H and A 1 C n H be given, and assume that the matrix equation AXA = B has a solution for X C n H and define S = { X Cn H AXA = B}. Then, Hence, max i ±( X A 1 ) = n + i ± ( B AA 1 A ) r(a), (2.25) min i ±( X A 1 ) = i ± ( B AA 1 A ), (2.26) max i ±(X) = n + i ± (B) r(a), (2.27) min i ±(X) = i ± (B). (2.28) 7

(a) AXA = B has a solution X > A 1 (X < A 1 ) if and only if i + ( B AA 1 A ) = r(a) (i ( B AA 1 A ) = r(a)). (b) AXA = B has a solution X A 1 (X A 1 ) if and only if B AA 1 A (B AA 1 A ). (c) AXA = B has a solution X > (X < ) if and only if B and r(a) = r(b) (B and r(a) = r(b)). (d) AXA = B has a solution X (X ) if and only if B (B ). The global maximum and minimum ranks and inertias of A B 1 XB 1 subject to a pair of matrix equations We first derive a parametric form for the general common Hermitian solution of the pair of matrix equations in (1.2) (1.5). Lemma.1 (1) Let A i C mi H, B i C mi n be given for i = 2,, and suppose that each of the two matrix equations has a solution, i.e., R(A i ) R(B i ) for i = 2,. Then, B 2 XB 2 = A 2 and B XB = A (.1) (a) The pair of matrix equations have a common Hermitian solution if and only if A B 2 r A B B2 = 2r. (.2) B2 B B (b) Under (.2), the general common Hermitian solution of the pair of equations can be written in the following parametric form X = X + V F B + F B V + F B2 UF B + F B U F B2, (.) B2 where X is a special Hermitian common solution to the pair of equations, B =, and U, V C n n are arbitrary. Substituting (.) into A 1 B 1 XB 1 gives A 1 B 1 XB 1 = A 1 B 1 X B 1 B 1 V F B B 1 B 1 F B V B 1 B 1 F B2 UF B B 1 B 1 F B U F B2 B 1, (.4) which is a matrix expression involving two variable matrices V and U. Thus, the constrained matrix expression in (1.2) is equivalently converted to the unconstrained matrix expression in (.4). To find the global maximum and minimum ranks and inertias of (.4), we need the following result. Lemma.2 Let p(x 1, X 2 ) = A B 1 X 1 C 1 (B 1 X 1 C 1 ) B 2 X 2 C 2 (B 2 X 2 C 2 ), (.5) where A C m H, B i C m pi and C i C qi m are given, and X i C pi qi are variable matrices for i = 1, 2, and assume that Also let N = M = A B2 C 1 C 2 C 1 R(B 2 ) R(B 1 ), R(C 1 ) R(B 1 ), R(C 2 ) R(B 1 ). (.6) A, M C 1 1 = A B 2 C1 C2 A B 2 C, N 1 = B2 1 C2, N 2 = C 1, C 1 C A B 2 C1 A C B2 1 C2, M 2 = C 1. C 1 C B 8

Then, the global maximum and minimum ranks and inertias of p(x 1, X 2 ) are given by max r p(x 1, X 2 ) = min{r A, B 1, r(n), r(m 1 ), r(m 2 )}, (.7) X 1 C p 1 q 1, X 2 C p 2 q 2 min r p(x 1, X 2 ) = 2r A, B 1 2r(M) + 2r(N) + max{ s 1, s 2, s, s 4 }, (.8) X 1 C p 1 q 1, X 2 C p 2 q 2 max i ± p(x 1, X 2 ) = min{i ± (M 1 ), i ± (M 2 )}, (.9) X 1 C p 1 q 1, X 2 C p 2 q 2 min i ± p(x 1, X 2 ) = r A, B 1 r(m) + r(n) X 1 C p 1 q 1, X 2 C p 2 q 2 where + max{ i ± (M 1 ) r(n 1 ), i ± (M 2 ) r(n 2 ) }, (.1) s 1 = r(m 1 ) 2r(N 1 ), s 2 = r(m 2 ) 2r(N 2 ), s = i + (M 1 ) + i (M 2 ) r(n 1 ) r(n 2 ), s 4 = i + (M 1 ) + i (M 2 ) r(n 1 ) r(n 2 ). Proof Under (.6), applying Lemma 1.7 to the variable matrix X 1 in (.5) and simplifying, we obtain { } max r p(x 1, X 2 ) = min r A B 2 X 2 C 2 (B 2 X 2 C 2 ) A B2 X, B 1, r 2 C 2 (B 2 X 2 C 2 ) C1 X 1 C 1 { } A B2 X = min r A, B 1, r 2 C 2 (B 2 X 2 C 2 ) C1, (.11) C 1 min r p(x 1, X 2 ) = 2r A B 2 X 2 C 2 (B 2 X 2 C 2 ) A B2 X, B 1 + r 2 C 2 (B 2 X 2 C 2 ) C1 X 1 C 1 A B2 X max i ± p(x 1, X 2 ) = i 2 C 2 (B 2 X 2 C 2 ) C1 ±, (.12) X 1 C 1 A B2 X 2r 2 C 2 (B 2 X 2 C 2 ) B 1 C 1 A B2 X = 2r A, B 1 + r 2 C 2 (B 2 X 2 C 2 ) C1 A 2r, (.1) C 1 C 1 min i ± p(x 1, X 2 ) = r A B 2 X 2 C 2 (B 2 X 2 C 2 ) A B2 X, B 1 + i 2 C 2 (B 2 X 2 C 2 ) C1 ± X 1 C 1 A B2 X r 2 C 2 (B 2 X 2 C 2 ) B 1 C 1 = r A, B 1 + i ± A B2 X 2 C 2 (B 2 X 2 C 2 ) C 1 C 1 Notice that A B2 X 2 C 2 (B 2 X 2 C 2 ) C1 A C = 1 C 1 C 1 Applying Lemma 1.6 to this expression gives r A C 1. (.14) B2 C X 2 C 2, 2 X 2 B2, := q(x 2 ). (.15) max rq(x 2 ) = min { r(n), r(m 1 ), r(m 2 ) }, (.16) X 2 C m p 2 min rq(x 2 ) = 2r(N) + max{ s 1, s 2, s, s 4 }, (.17) X 2 C m p 2 max i ± q(x 2 ) = min{ i ± (M 1 ), i ± (M 2 ) }, (.18) X 2 C m p 2 min i ± q(x 2 ) = r(n) + max{ i ± (M 1 ) r(n 1 ), i ± (M 2 ) r(n 2 ) }, (.19) X 2 C m p 2 where s 1 = r(m 1 ) 2r(N 1 ), s 2 = r(m 2 ) 2r(N 2 ), 9

s = i + (M 1 ) + i (M 2 ) r(n 1 ) r(n 2 ), s 4 = i (M 1 ) + i + (M 2 ) r(n 1 ) r(n 2 ). Substituting these results into (.11) (.14) yields (.9) (.2). It is obviously of great importance to be able to give analytical formulas for calculating the global maximum and minimum ranks and inertias of the matrix expression in (.5) under the assumptions in (.7). However, it is not easy to find the global maximum and minimum ranks and inertias ranks and inertias of a general p(x 1, X 2 ) as given in (.5). For convenience of representation, we rewrite (.4) as where A 1 B 1 XB 1 = A G 1 V G 2 (G 1 V G 2 ) G UG 4 (G UG 4 ), (.2) A = A 1 B 1 X B 1, G 1 = B 1, G 2 = F B B 1, G = B 1 F B2, G 4 = F B B 1. (.21) It is easy to verify that the above matrices satisfy the conditions (G 2) R(G 1 ), R(G ) R(G 1 ), R(G 4) R(G 1 ), R(G 2) R(G ), R(G 2) R(G 4). (.22) In this case, applying Lemma.2 to (.22) yields the main results of this section. Theorem. Let A i C mi H equations and B i C mi n be given for i = 1, 2,, and assume that the pair of matrix B 2 XB 2 = A 2 and B XB = A (.2) have a common solution X C n H. Also denote the set of all their common Hermitian solutions by and let A1 B P 1 = 1 B2 B S = {X C n H B 2 XB 2 = A 2, B XB = A }. (.24), P 2 = A 1 B 1 B 1 A 1 B 1 B 1 Q 1 = A B A B, Q 2 = A 2 B B B2 B 1 B2 B Then, (a) The global maximum rank of A 1 B 1 XB 1 subject to (.24) is max r( A 1 B 1 X ) { = min r A 1, B 1, r(q 1 ) r A 1 B 1 A 2 B 2, P = A 1 B 1 A B, (.25) B2 B A 1 B 1 B 1, Q = A B B. B 2 (.26) B2 (b) The global minimum rank of A 1 B 1 XB 1 subject to (.24) is where min r( A 1 B 1 X ) } r(b B 2 ) r(b ), r(p 2 ) 2r(B 2 ), r(p ) 2r(B ). (.27) = 2r A 1, B 1 2r(P 1 ) + 2r(Q 1 ) + max{ r(p 2 ) 2r(Q 2 ), r(p ) 2r(Q ), u 1, u 2 }, (.28) u 1 = i + (P 2 ) + i (P ) r(q 2 ) r(q ), u 2 = i (P 2 ) + i + (P ) r(q 2 ) r(q ). (c) The global maximum partial inertia of A 1 B 1 XB 1 subject to (.24) is max i ±( A 1 B 1 X ) = min {i ± (P 2 ) r(b 2 ), i ± (P ) r(b )}. (.29) 1

(d) The global minimum partial inertia of A 1 B 1 XB 1 subject to (.24) is min i ±( A 1 B 1 X ) = r A 1, B 1 r(p 1 ) + r(q 1 ) (.) + max{i ± (P 2 ) r(q 2 ), i ± (P ) r(q )}. (.1) Proof Under (.22), we find by Lemma.2 that max r( A 1 B 1 X ) = max r A G 1V G 2 (G 1 V G 2 ) G UG 4 (G UG 4 ) V, U { A G G = min r A, G 1, r 4 A G, r G G, r A G 4 G 4 min r( A 1 B 1 X ) = min r A G 1V G 2 (G 1 V G 2 ) G UG 4 (G UG 4 ) V, U A G1 A G G = 2r A, G 1 2r + 2r 4 G G }, (.2) + max{s 1, s 2, s, s 4 }, (.) max i ±( A 1 B 1 X ) = max i ± A G 1 V G 2 (G 1 V G 2 ) G UG 4 (G UG 4 ) V, U { } A G A G = min i ± G, i 4 ±, (.4) G 4 min i ±( A 1 B 1 X ) = min i ± A G 1 V G 2 (G 1 V G 2 ) G UG 4 (G UG 4 ) V, U A G1 A G G = r A, G 1 r + r 4 G G + max{t 1, t 2 }, (.5) where A G A G G s 1 = r G 2r 4 G, A G s 2 = r 4 A G G 2r 4, G 4 G 4 A G A G s = i + G + i 4 A G G r 4 G 4 G A G A G s 4 = i G + i 4 A G G + r 4 G 4 G A G A G G t 1 = i ± G r 4 G, A G4 A G G t 2 = i ± G r 4 4 G. 4 A G G r 4, G 4 r A G G 4 G 4, 11

Applying (1.1) (1.12) and (1.19), and simplifying by ( B 2 X B 2, B X B ) = ( A 2, A ), elementary matrix operations and congruence matrix operations, we obtain r A, G 1 = r A 1 B 1 X, B 1 = r A 1, B 1, (.6) A G G r 4 A1 B = r 1 X B 1 F B2 B 1 F B G F B A 1 B 1 X B 1 B 1 = r B B r(b) r(b 2) r(b ) B A 1 B 1 B 1 B 1 X B = r B B r(b) r(b 2) r(b ) B A 1 B 1 B 1 = r B2 B B A r(b) r(b 2) r(b ) B A A G1 A1 B r = r 1 X B 1 G F B G A i ± G = r(q 1 ) r(b) r(b 2 ) r(b ), (.7) A1 B = r 1 B r(b) = r(p 1 ) r(b), (.8) A1 B = i 1 X A B 1 F 1 B 1 X B 1 B2 ± F B2 = i ± B2 r(b 2 ) B A 1 B 1 B 1 X B2/2 A 1 B 1 = i ± B2 r(b 2 ) = i ± B B 1 X B2/2 1 B 2 r(b 2 ) B B 2 A 2 = i ± (P 2 ) r(b 2 ), (.9) A G G r 4 A1 B G = r 1 X B 1 F B2 B 1 F B F B2 A 1 B 1 X B 1 B 1 = r B2 B 2r(B 2) r(b ) B A 1 B 1 B 1 B 1 X B2 = r B 2 B 2r(B 2) r(b ) B A 1 B 1 B 1 = r B2 2r(B 2) r(b ) B A 2 B = r(q 2 ) 2r(B 2 ) r(b ). (.4) By a similar approach, we can obtain A G4 A G G i ± G = i 4 ± (P ) r(b ), r 4 = r(q G 4 ) r(b 2 ) 2r(B ). (.41) Substituting (.6) (.41) into (.2) (.5) yields (.27) (.1). Some direct consequences of the previous theorem are given below. 12

Corollary.4 Let A i C mi H and B i C mi n be given for i = 1, 2,, and suppose that each pair of B 1 X = A 1, B 2 XB2 = A 2 and B XB = A have a common Hermitian solution. Also let S be of the form (.2). Then, { max r( A 1 B 1 X B2 ) = min r(b 1 ), r(q 1 ) r r(b B 2 ) r(b ), } 2r 2r(B B 2 ), 2r 2r(B 2 B ), (.42) B 1 B 1 B 1 min r( A 1 B 1 X ) = 2r(Q 1 ) 2r B 2 2r B, (.4) B B { } max i ±( A 1 B 1 X ) = min r r(b B 2 ), r r(b 2 B ), (.44) min i ±( A 1 B 1 X ) = r(q 1 ) r B 1 B 2 r B 1 B 1 B, (.45) B B where Q 1 is of the form (.26). Proof Under the given conditions, the ranks/inertias of the block matrices in (.25) and (.26) are given by r(p 1 ) = r(b 1 ) + r, r(p ) = 2r, i ± (P 2 ) = r, i ± (P ) = r, B 1 B 2 B, r(p 2 ) = 2r B 2 B 1 B 1 r(q 2 ) = r B B Hence (.27) (.1) reduce to (.42) (.45). + r B 2 B B 1 B 1, r(q ) = r B B B 2 + r B Corollary.5 Let A i C mi mi H and B i C mi n be given for i = 1, 2,, and suppose that each pair of the triple matrix equations B 1 XB 1 = A 1, B 2 XB 2 = A 2, B XB = A (.46) have a common Hermitian solution. Then, there exists a Hermitian X such that (.46) holds if and only if A 1 B 1 B 1 r A B B 1 B 1 A B = r B + r, B2, B. (.47) B2 B B Proof It follows from (.4). A challenging open problem on the triple matrix equations in (.46) is to give a parametric form for their general common Hermitian solution. Setting B 1 = I n in Theorem. may yield a group of results on the extremal ranks/inertias of A 1 X subject to (.24). In particular, we have the following consequences. Corollary.6 Let A i C mi H and B i C mi n be given for i = 2,, and assume that (.2) has a common solution. Also let S be of the form (.24). Then, (a) The global maximum rank of the solution of (.24) is where max r(x) = min{ n, s 1, s 2, s }, (.48) A B s 1 = 2n + r 2 B2 r r(b A B B 2 ) r(b ), s 2 = 2n + r(a 2 ) 2r(B 2 ), s = 2n + r(a ) 2r(B ).. B 1

(b) The global minimum rank of the solution of (.24) is A B min r(x) = 2r 2 + max{ t A B 1, t 2, t, t 4 }, (.49) where A2 B t 1 = r(a 2 ) 2r 2 B t = i + (A 2 ) + i (A ) r t 4 = i (A 2 ) + i + (A ) r B2, t 2 = r(a ) 2r A2 B B2 r, B A B A2 B B2 B r A B A B,. (c) The global maximum partial inertia of the solution of (.24) is max i ±(X) = min{ n + i ± (A 2 ) r(b 2 ), n + i ± (A ) r(b ) }. (.5) (d) The global minimum partial inertia of the solution of (.24) is { min i A B ±(X) = r 2 A2 B + max i A B ± (A 2 ) r 2 B Hence, (e) Eq. (.2) has a solution X > if and only if, i ± (A ) r A 2, A, R(A 2 ) = R(B 2 ), R(A ) = R(B ). (f) All solutions of (.2) satisfy X > if and only if A 2, A and one of (g) Eq. (.2) has a solution X < if and only if r(a 2 ) = r(b 2 ) = n, r(a ) = r(b ) = n. A 2, A, R(A 2 ) = R(B 2 ), R(A ) = R(B ). (h) All solutions of (.2) satisfy X < if and only if A 2, A and one of r(a 2 ) = r(b 2 ) = n, r(a ) = r(b ) = n. (i) Eq. (.2) has a solution X if and only if A B2 A 2, A, R R A B, R R A (j) All solutions of (.2) satisfy X if and only if A 2, A and one of r(b 2 ) = n and r(b ) = n. B2 A2 B 2 A B B (k) Eq. (.2) has a solution X if and only if A B A2 B A 2, A, R R, R R 2. A B A B (l) All solutions of (.2) satisfy X if and only if A 2, A and one of r(b 2 ) = n and r(b ) = n.. }. (.51) 14

Proof Set A 1 = and B 1 = I n in Theorem. and simplifying, we obtain (a) (d). Applying Lemma 1.2 to (.49) and (.5), we obtain (e) (l). Rewrite B 2 XB2 = A 2 and B XB = A as X1 X B 21, B 22 2 B 21 X1 X X2 = A X 2, B 1, B 2 2 B 1 X2 = A X, (.52) B 22 where B i1 C mi n1, B i2 C mi n2, i = 2,, X 1 C n1 H, X 2 C n1 n2 and X C n2 H with n 1 + n 2 = n. We next derive the extremal ranks and inertias of the submatrices X 1 and X in a Hermitian solution of (.52). Note that X 1, X 2, X in (.52) can be rewritten as X 1 = P 1 XP 1, X 2 = P 1 XP 2, X = P 2 XP 2, (.5) where P 1 = I n1, and P 2 =, I n2. For convenience, we adopt the following notation for the collections of the submatrices X 1 and X in (.52): S 1 = {X 1 = P 1 XP 1 B 2 XB 2 = A 2, B XB = A, X = X }, (.54) S = {X = P 2 XP 2 B 2 XB 2 = A 2, B XB = A, X = X }. (.55) The maximum and minimum ranks and inertias of the submatrices X 1 and X in (.52) can easily be derived from Theorem.. The details are omitted. If each of the triple matrix equations in (1.6) is not consistent, people may alternatively seek its common approximation solutions under various given optimal criteria. One of the most useful approximation solutions of BXB = A is the least-squares Hermitian solution, which is defined to be a Hermitian matrix X that minimizes the norm of the difference A BXB : A BXB 2 = tr ( A BXB )( A BXB ) = min. (.56) The normal equation corresponding to the norm minimization problem is given by B 2 B BXB B = B AB. (.57) This equation is always consistent. Concerning the common least-squares Hermitian solution of (1.6), we have the following result. Corollary.7 Let A i C mi mi H and B i C mi n be given for i = 1, 2,. Then, triple matrix equations have a common least-squares Hermitian solution, namely, there exists an X C n n H such that if and only if r r B i A ib i B i B i B j A jb j B j B j B i B i B j B j A i B i XB i = min, i = 1, 2,, (.58) A 1 B 1 B 1 B 1 B2A 2 B B2B BA B BB B 1 B2B 2 BB Proof It follows from Lemma.1, Corollary.5 and (.57). = 2r Bi B j, i j, i, j = 1, 2,, (.59) = r B 1 B 1 B + r B 1 B 2. (.6) B B Some further research problems can be proposed. For example, a challenging task is to give the closed-form for the general common solution of B 2 XB2 = A 2 and B XB = A that satisfies X > (<,, ), which is equivalent to solving the inequalities X + V F B + F B V + F B2 UF B + F B U F B2 > (<,, ). (.61) Further, it would be of interest to consider the following optimization problems: (a) Maximize and minimize the rank and inertia of the Hermitian matrix expression A 1 B 1 XB 1 subject to k 1 consistent Hermitian matrix equations ( B 2 XB 2,..., B k XB k ) = ( A 2,..., A k ), and to give necessary and sufficient condition for the set of matrix equations ( B 1 XB 1,..., B k XB k ) = ( A 1,..., A k ) to have a common Hermitian solution. (b) Maximize and minimize the rank and inertia of the Hermitian matrix expression A 1 B 1 XB 1 subject to the matrix inequality B 2 XB 2 A 2. 15

4 Conclusions In this paper, we studied the problems of maximizing/minimizing the rank/inertia of the constrained matrix expression in (1.2), and obtained some closed-form formulas for the extremal ranks/inertias of (1.2) by pure algebraic operations of matrices and their generalized inverses. As direct applications, we gave necessary and sufficient conditions for the existence of X satisfying the triple matrix equations in (1.6), as well as some matrix inequalities. Although the problems of maximizing/minimizing the rank/inertia are regarded as NP-hard, the results obtained in this paper and 12, 14, 15, 16, 17, 28, 29,, 2, show that closed-form formulas for the extremal ranks/inertias of some simpler matrix expressions can be derived, while these closed-form formulas can be used to solve some fundamental problems in matrix theory, as mentioned in the beginning of this paper. All the results obtained in these papers are brandnew and beyond our conventional understanding on linear matrix expressions. This series of researches show that for many basic or classic problems like solvability of matrix equations and matrix inequalities, we are still able to develop some new methods and use them to derive a variety of innovative results. References 1 E. Candes and B. Recht, Exact matrix completion via convex optimization, Found. of Comput. Math. 9, 717 772, 29. 2 M. Fazel, H. Hindi and S. Boyd, A Rank minimization heuristic with application to minimum order system approximation, In: Proceedings of the 21 American Control Conference, pp. 474 479, 21. M. Fazel, H. Hindi and S. Boyd, Rank minimization and applications in system theory, In: Proceedings of the 24 American Control Conference pp. 27 278, 24. 4 J.F. Geelen, Maximum rank matrix completion, Linear Algebra Appl. 288(1999), 211 217. 5 J. Groß, A note on the general Hermitian solution to AXA = B, Bull. Malays. Math. Soc. (2) 21(1998), 57 62. 6 N.J.A. Harvey, D.R. Karger and S. Yekhanin, The complexity of matrix completion, In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, Association for Computing Machinery, New York, pp. 11 1111, 26. 7 T.M. Hoang and T. Thierauf, The complexity of the inertia, Lecture Notes in Computer Science 2556, Springer, 22, pp. 26 217. 8 T.M. Hoang and T. Thierauf, The complexity of the inertia and some closure properties of GapL, In: Proceedings of the Twentieth Annual IEEE Conference on Computational Complexity, 25, pp. 28 7. 9 C.G. Khatri and S.K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math. 1(1976), 579 585. 1 Y. Kim and M. Mesbahi, On the rank minimization problem, In: Proceedings of the 24 American Control Conference, Boston, pp. 215 22, 24. 11 M. Laurent, Matrix completion problems, In: Encyclopedia of Optimization (C.A. Floudas and P.M. Pardalos, eds.), Vol. III, Kluwer, pp. 221 229, 21. 12 Ying Li, F. Zhang, W. Guo and J. Zhao, Solutions with special structure to the linear matrix equation AX = B, Comput. Math. Appl. 61(211), 74 8. 1 X. Liu and J. Rong, On Hermitian nonnegative-definite solutions to matrix equations, Math. Notes 85(29) 45 457. 14 Y. Liu and Y. Tian, More on extremal ranks of the matrix expressions A BX ± X B with statistical applications, Numer. Linear Algebra Appl. 15(28), 7 25. 15 Y. Liu and Y. Tian, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA = B with applications, J. Appl. Math. Comput. 2(21), 289 1. 16 Y. Liu and Y. Tian, A simultaneous decomposition of a matrix triplet with applications, Numer. Linear Algebra Appl. 18(211), 69 85. 17 Y. Liu and Y. Tian, Max-min problems on the ranks and inertias of the matrix expressions A BXC ± (BXC) with applications, J. Optim. Theory Appl. 148(211), 59 622. 18 Y. Liu, Y. Tian and Y. Takane, Ranks of Hermitian and skew-hermitian solutions to the matrix equation AXA = B, Linear Algebra Appl. 41(29), 259 272. 19 G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra 2(1974), 269 292. 2 M. Mahajan and J. Sarma, On the complexity of matrix rank and rigidity, Lecture Notes in Computer Science Vol. 4649, Springer, pp. 269 28, 27. 16

21 M. Mesbahi, On the rank minimization problem and its control applications, Systems & Control Letters (1998), 1 6. 22 M. Mesbahi and G.P. Papavassilopoulos, Solving a class of rank minimization problems via semi-definite programs, with applications to the fixed order output feedback synthesis, In: Proceedings of the American Control Conference, Albuquerque, New Mexico, pp. 77 8, 1997. 2 B.K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput. 24(1995), 227 24. 24 A B. Özgüler and N. Akar, A common solution to a pair of linear matrix equations over a principal ideal domain, Linear Algebra Appl. 144(1991), 85 99. 25 B. Recht, M. Fazel and P.A. Parrilo, Guaranteed minimum rank solutions to linear matrix equations via nuclear norm minimization, SIAM Review 52(21), 471 51. 26 G.A.F. Seber, A Matrix Handbook for Statisticians, John Wiley & Sons, 28. 27 Y. Tian, Solvability of two linear matrix equations, Linear and Multilinear Algebra 48(2), 12 147. 28 Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl. 4(21), 26 296. 29 Y. Tian, Rank and inertia of submatrices of the Moore Penrose inverse of a Hermitian matrix, Electron. J. Linear Algebra 2(21), 226 24. Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias. Electron. J. Linear Algebra 21(21), 124 141. 1 Y. Tian, Maximization and minimization of the rank and inertia of the Hermitian matrix expression A BX (BX) with applications, Linear Algebra Appl. 44(211), 219 219. 2 Y. Tian, On additive decompositions of the Hermitian solutions of the matrix equation AXA = B, Mediter. J. Math., DOI:1.17/s9-1-11-8. Y. Tian and Y. Liu, Extremal ranks of some symmetric matrix expressions with applications, SIAM J. Matrix Anal. Appl. 28(26), 89 95. 4 J. Wang, V. Sreeram and W. Liu, The parametrization of the pencil A + BKC with constant rank and its application, Internat. J. Infom. Sys. Sci. 4(28), 488 499. 5 X. Zhang, The general common Hermitian nonnegative-definite solution to the matrix equations AXA = BB and CXC = DD with applications in statistics, J. Multivariate Anal. 9(25), 257 266. 17