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Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela RANK AND INERTIA OF SUBMATRICES OF THE MOORE PENROSE INVERSE OF A HERMITIAN MATRIX YONGGE TIAN Abstract. Closed-form formulas are derived for te rank and inertia of submatrices of te Moore Penrose inverse of a Hermitian matrix. A variety of consequences on te nonsingularity, nullity and definiteness of te submatrices are also presented. Key words. Hermitian matrix, Partitioned matrix, Moore Penrose inverse, Rank, Inertia. AMS subject classifications. 15A, 15A9, 15A2, 15A57. 1. Introduction. Trougout tis paper, C m n and C m m stand for te sets of all m n complex matrices and all m m Hermitian complex matrices, respectively. Te symbols A, r(a) and R(A) stand for te conjugate transpose, rank and range (column space) of a matrix A C m n, respectively. [A, B denotes a row block matrix consisting of A and B. Te inertia of a Hermitian matrix A is defined to be te triplet In(A) = {i + (A), i (A), i (A)}, were i + (A), i (A) and i (A) are te numbers of te positive, negative and zero eigenvalues of A counted wit multiplicities, respectively. It is obvious tat r(a) = i + (A) + i (A). We write A > (A ) if A is Hermitian positive (nonnegative) definite. Two Hermitian matrices A and B of te same size are said to satisfy te inequality A > B (A B) in te Löwner partial ordering if A B is positive (nonnegative) definite. Te Moore Penrose inverse of A C m n, denoted by A, is defined to be te unique solution X of te four matrix equations (i) AXA = A, (ii) XAX = X, (iii) (AX) = AX, (iv) (XA) = XA. Amatrix X C m m is calledahermitian g-inverseofa, denoted bya, ifit satisfies AXA = A. Furter, te symbols E A and F A stand for te two ortogonal projectors (idempotent Hermitian matrices) E A = I m AA and F A = I n A A. A well-known property of te Moore Penrose inverse is (A ) = (A ). In particular, (A ) = A and AA = A A if A = A. Results on te Moore Penrose inverse can be found, e.g., in [1, 2, 7. Received by te editors September, 29. Accepted for publication April 19, 21. Handling Editor: Mose Goldberg. Cina Economics and Management Academy, Central University of Finance and Economics, Beijing 181, Cina (yongge.tian@gmail.com) 226

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 227 One of te fundamental operations in matrix teory is to partition a matrix into block forms. Many properties of a matrix and its operations can be derived from partitions of te matrix and teir operations. A typical partitioned Hermitian matrix is given by te following 2 2 form [ A B M = B, (1.1) D were A C m m, B C m n and D C n n. Correspondingly, te Moore Penrose inverse of M is Hermitian as well, and a partitioned expression of M can be written as [ M G1 G = 2 G 2 G, (1.2) were G 1 C m m, G 2 C m n and G C n n. Wen M in (1.1) is nonsingular, (1.2) reduces to te usual inverse of M. In te investigation of a partitioned matrix and its inverse or generalized inverse, attention is often given to expressions of submatrices of te inverse or generalized inverse, as well as teir properties. If (1.1) is nonsingular, explicit expressions of te tree submatrices G 1, G 2 and G of te inverse in (1.2) were given in a recent paper [19. If (1.1) is singular, expressions of te tree submatrices G 1, G 2 and G in (1.2) can also be derived from certain decompositions of M. Various formulas for G 1, G 2 and G in (1.2) were given in te literature; see, e.g., [9, 1, 16, 17. Tese expressions, owever, are quite complicated in general. In addition to te expressions of te submatrices in (1.2), anoter important task is to describe various algebraic properties of te submatrices in (1.2), suc as, teir rank, range, nullity, inertia, and definiteness. Some previous and recent work on tese properties can be found, e.g., in [4, 1, 11, 14, 15, 18, 19, 2, 21. Motivated by te work on nullity and inertia of submatrices in a nonsingular (Hermitian) matrix and its inverse, we derive in tis paper closed-form formulas for te rank and inertia of te submatrices G 1, G 2 and G in (1.2) troug some known and new results on ranks and inertias of (Hermitian) matrices. As applications, we use tese formulas to caracterize te nonsingularity, nullity and definiteness of te submatrices in (1.2). Some well-known equalities and inequalities for ranks of partitioned matrices are given below. Lemma 1.1 ([12). Let A C m n, B C m k, C C l n and D C l k. Ten,

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela 228 Y. Tian (a) Te following rank equalities old r[a, B = r(a)+r(e A B) = r(b)+r(e B A), (1.) [ A r = r(a)+r(cf A ) = r(c)+r(af C ), (1.4) C [ A B r = r(b)+r(c)+r(e B AF C ), (1.5) C [ A B r B = r[a, B+r(B) if A, (1.6) [ [ A B EA B r = r(a)+r C D CF A D CA. (1.7) B (b) Te following rank inequalities old [ A B r(a)+r(b)+r(c) r r(b)+r(c), (1.8) C [ [ A B A r(ca B) r r r[a, B+r(A). (1.9) C C [ A B (c) r = r(a)+r(b)+r(c) R(A) R(B) = {} and R(A ) R(C ) = C {}. [ A B (d) r = r(a) AA B = B, CA A = C and D = CA B. C D Note tat te inertia of a Hermitian matrix divides te eigenvalues of te matrix into tree sets on te real line. Hence te inertia of a Hermitian matrix can be used to caracterize te definiteness of te matrix. Te following result is obvious from te definitions of te rank and inertia of a matrix. Lemma 1.2. Let A C m m, B C m n and C C m m. Ten, (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) = ).

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 229 Lemma 1.. Let A C m m, B C m n and P C m m. Ten, i ± (PAP ) = i ± (A) if P is nonsingular, (1.1) i ± (PAP ) i ± (A) if P is singular, (1.11) A(A ) A = A, r(a ) = r(a), (1.12) i ± (A ) = i ± (A), i ± (A ) = i ± (A), (1.1) { i± (A) if λ > i ± (λa) = i (A) if λ <, (1.14) [ B i ± B = r(b). (1.15) Equation (1.1) is te well-known Sylvester s law of inertia (see, e.g., [8, Teorem 4.5.8). Equation (1.11) was given in [18, Lemma 1.6. Equations (1.12), (1.1) and (1.14) follow from te spectral decomposition of A and te definitions of te Moore Penrose inverse, rank and inertia. Equation (1.15) is well known (see, e.g., [5, 6). Ten, Te following result was given in [18, Teorem 2.. Lemma 1.4. Let A C m m In particular, U = [ A B B, V =, B C m n, D C n n, and denote [ A B B, S A = D B A B. D i ± (U) = r(b)+i ± (E B AE B ), (1.16) [ E A B i ± (V) = i ± (A)+i ± B, (1.17) E A S A r(b) i ± (U) r(b)+i ± (A), (1.18) r[a, B i (A) i ± (V) r[a, B+i ± (S A ) i (A), (1.19) i ± (B A B) i (U) r[a, B+i ± (A). (1.2) (a) If A, ten i + (U) = r[a, B and i (U) = r(b). (b) If R(A) R(B), ten i ± (U) = r(b). (c) If R(A) R(B) = {}, ten i ± (U) = i ± (A)+r(B). (d) If R(A) R(B) = {} and R(B ) R(D) = {}, ten i ± (V) = i ± (A) + r(b)+i ± (D). (e) i ± (V) = i ± (A) if and only if R(B) R(A) and i ± (D B A B) =.

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela 2 Y. Tian Lemma 1.5. Let A C m m and B C m n. Ten, [ A i ± (B A AB B) = i (AB) [ A r(b A AB B) = r (AB) i (A), (1.21) r(a). (1.22) Proof. Applying (1.12), (1.14) and (1.17) gives [ A AB i ± (AB) = i ± (A )+i ± [ B A(A ) AB = i ± (A)+i (B A B), as required for (1.21). Adding te two equalities in (1.21) gives (1.22). 2. Main results. Note tat te tree submatrices G 1, G 2 and G in (1.2) can be represented as G 1 = P 1 M P 1, G 2 = P 1 M P 2, G = P 2 M P 2, (2.1) were P 1 = [I m, and P 2 = [, I n. Applying Lemma 1.5 to (2.1) gives te following result. Teorem 2.1. Let M and M be given by (1.1) and (1.2), and denote W 1 = [A, B and W 2 = [B, D. (2.2) Ten, and r(g 1 ) = r 2 DW 2 W 1 r(g 2 ) = r 1 BW 2 W 2 r(g ) = r 1 AW 1 W 2 i ± (G 1 ) = i 2 DW 2 W 1 i ± (G ) = i 1 AW 1 W 2 r(m), (2.) r(m), (2.4) r(m), (2.5) i (M), (2.6) i (M). (2.7) Hence,

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 21 W (a) G 1 is nonsingular if and only if r[ 2 DW 2 W1 = r(m)+m. W (b) G 1 = if and only if r[ 2 DW 2 W1 = r(m). (c) G 1 > (G 1 < ) if and only if i 2 DW 2 W 1 = i (M)+m (d) G 1 (G 1 ) if and only if i + 2 DW 2 W 1 = i + (M) ( i 2 DW 2 W ) 1 + = i + (M)+m. ( i 2 DW 2 W ) 1 = i (M). W (e) G 2 = if and only if r[ 1 BW 2 W2 = r(m). W (f) G is nonsingular if and only if r[ 1 AW 1 W2 = r(m)+m. W (g) G = if and only if r[ 1 AW 1 W2 = r(m). () G > (G < ) if and only if i 1 AW 1 W 2 = i (M)+m ( i 1 AW 1 W ) 2 + = i + (M)+m. (i) G (G ) if and only if i + 1 AW 1 W 2 = i + (M) ( i 1 AW 1 W ) 2 = i (M). Proof. Applying (1.21) and (1.22) to (2.1) gives r(g 1 ) = r[ M MP 1 P 1 M r(g 2 ) = r[ M MP 2 P 1 M r(g ) = r[ M MP 2 P 2 M i ± (G 1 ) = i [ M MP 1 P 1 M i ± (G ) = i [ M MP 2 P 2 M M W1 r(m) = r[ M W2 r(m) = r[ M W2 r(m) = r[ [ M W1 i (M) = i i (M) = i [ M W 2 r(m), (2.8) r(m), (2.9) r(m), (2.1) i (M), (2.11) i (M). (2.12)

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela 22 Y. Tian Expanding M gives M = [W 1, W 2 [ A B B D Hence, [ Im 1 2 W 1A W 2B = [ W1 W 2 = W1 AW 1 +W1 BW 2 +W2 B W 1 +W2 DW 2. [ M W 1 [ I m 1 2 AW 1 BW 2 = I n [ M W1 AW 1 W1 BW 2 W2 B W 1 W1 Applying (1.1) to tese equalities gives I n 2 DW 2 W 1. r[ M W 1 = r 2 DW 2 W 1 and i [ M W 1 = i 2 DW 2 W 1. Substituting tese equalities into (2.8) and (2.11) leads to (2.) and (2.6). Equations (2.5) and (2.7) can be sown similarly. Note tat [ Im W 1 A W 2 B = [ M W 2 [ I m DW 2 I n = I n [ M W1 AW 1 W2 B W 1 W2 DW 2 W1 1 BW 2 W2. Hence, r[ M W 2 = r 1 BW 2 W 2. Substituting tis equality into (2.9) leads to (2.4). Results (a) (i) follow from (2.) (2.7) and Lemma 1.2. We next obtain some consequences of Teorem 2.1 under various assumptions for M in (1.1). Corollary 2.2. Let M and M be given by (1.1) and (1.2), and assume tat M satisfies te rank additivity condition r(m) = r[a, B+r[B, D, i.e., R([A, B ) R([B, D ) = {}. (2.1)

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 2 Ten, r(g 1 ) = r(d)+r[a, B r[b, D, (2.14) r(g 2 ) = r(b), (2.15) r(g ) = r(a)+r[b, D r[a, B, (2.16) r(g 1 )+r(g ) = r(a)+r(d), (2.17) i ± (G 1 ) = i (D)+r[A, B i (M), (2.18) i ± (G ) = i (A)+r[B, D i (M), (2.19) i ± (G 1 )+i (G ) = i ± (A)+i (D). (2.2) In particular, if M is nonsingular, ten r(g 1 ) = r(d)+m n, r(g 2 ) = r(b), r(g ) = r(a)+n m, (2.21) i ± (G 1 ) = i (D)+m i (M), i ± (G ) = i (A)+n i (M). (2.22) Proof. Under (2.1), it follows from Lemmas 1.1(c) and 1.4(c) tat W r[ 2 DW 2 W1 = r(w 2 DW 2 )+2r(W 1 ), (2.2) i 2 DW 2 W 1 = i (W 2DW 2 )+r(w 1 ), (2.24) [ BDB were te matrix W2DW BD 2 2 = D 2 B D satisfies [ [ [ [ Im BD BDB BD 2 I m I n D 2 B D D B = I n D. Hence, it follows from (1.1), (1.12) and (1.1) tat r(w 2DW 2 ) = r(d ) = r(d), i (W 2DW 2 ) = i (D ) = i (D). (2.25) Substituting (2.25) into (2.2) and (2.24), and (2.2) and (2.24) into (2.) and (2.6), leads to (2.14) and (2.18). Equations (2.15), (2.16) and (2.19) can be sown similarly. Adding (2.14) and (2.16) yields (2.17). Adding (2.18) and (2.19) yields (2.2). If M is nonsingular, ten r[a, B = m and r[b, D = n. Hence, (2.14) (2.2) reduce to (2.21) and (2.22). Tetreeformulasin(2.21)weregivenin[4inteformofnullityofmatrices,and te corresponding results are usually called te nullity teorem; see also [, 15, 2. Note tat i (A) = m i ± (A) i (A), i (D) = n i ± (D) i (D), i ± (M)+i (M) = m+n.

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela 24 Y. Tian Hence, (2.22) can alternatively be written as i ± (M) = i ± (D)+i (D)+i ± (G 1 ) and i ± (M) = i ± (A)+i (A)+i ± (G ). Tese formulas were given in [1, 11. Corollary 2.. Let M and M be given by (1.1) and (1.2), and assume tat M satisfies R(A) R(B) = {} and R(B ) R(D) = {}. (2.26) Ten, r(g 1 ) = r(a), r(g 2 ) = r(b), r(g ) = r(d), (2.27) i ± (G 1 ) = i ± (A), i ± (G ) = i ± (D). (2.28) Proof. Equation (2.26) is equivalent to r(w 1 ) = r(a)+r(b), r(w 2 ) = r(b)+r(d), r(m) = r(a)+2r(b)+r(d) (2.29) by(1.),(1.4) and(1.5). In tis case,(2.14) (2.16) reduce to(2.27). Also, substituting Lemma 1.4(d) and (2.29) into (2.18) and (2.19) yields (2.28). Corollary 2.4. Let M and M be given by (1.1) and (1.2), and assume tat r(m) = r(a). (2.) Ten, [ A B r(g 1 ) = r(a), r(g 2 ) = r(b), r(g ) = r B [ A B i ± (G 1 ) = i ± (A), i ± (G ) = i B r(a), (2.1) i (A). (2.2) Proof. By Lemma 1.1(d), (2.) is equivalent to E A B = and D = B A B, wic imply r(w 1 ) = r(a), r(w 2 ) = r(b), and R(W 2) R(W 1) by (1.). Applying

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 25 (1.5) to (2.) (2.5), and simplifying by elementary block matrix operations, produces r(g 1 ) = r 2 DW 2 W 1 r(g 2 ) = r 1 BW 2 W 2 r(g ) = r 1 AW 1 W 2 W r(m) = r[ 1 W r(m) = r[ 2 r(a) = r(a), r(a) = r(b), A A 2 B B r(m) = r B A 2 B AB D r(a) B D A A 2 B B = r r(a) B D A B = r r(a) B [ A B = r B r(a), as required for (2.1). Applying Lemma 1.4(b) and (e) to (2.6), and simplifying by elementary block congruence matrix operations and by (1.1), produces i ± (G 1 ) = i 2 DW 2 W 1 i ± (G ) = i 1 AW 1 W 2 i (M) = r(w 1 ) i (A) = i ± (A), A A 2 B B i (M) = i B A 2 B AB D i (A) B D A B = i i (A) B i (A), = i [ A B B as required for (2.2). Corollary 2.5. Let M and M be given by (1.1) and (1.2), and assume tat

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela 26 Y. Tian bot A and D. Ten, [ A BD r(g 1 ) = r B D 2 [ A 2 AB r(g ) = r B D [ A BD i + (G 1 ) = r[a, B i (M), i (G 1 ) = r B D 2 [ A i + (G ) = r[b 2 AB, D i (M), i (G ) = r B D Under te condition M, +r[a, B r(m), (2.) +r[b, D r(m), (2.4) i + (M), (2.5) i + (M). (2.6) r(g 1 ) = i + (G 1 ) = r(a) and r(g ) = i + (G ) = r(d). (2.7) Proof. If D, ten W2DW 2 and R(W2DW 2 ) = R(W2D). In tis case, applying (1.6) to (2.) gives W r(g 1 ) = r[ 2 DW 2 W1 r(m) = r[w2 DW, W 1 +r(w 1) r(m) = r[w2 D, W 1 +r(w 1) r(m) [ A BD = r B D 2 +r(w 1 ) r(m), as required for (2.). Equation (2.4) can be sown similarly. Applying Lemma 1.4(a) to (2.6) gives i + (G 1 ) = i 2 DW 2 W 1 i (M) = r(w 1 ) i (M) i (G 1 ) = i + 2 DW 2 W 1 = r[a, B i (M), i + (M) = r[w2 D, W 1 i +(M) [ A BD = r B D 2 i + (M), as required for (2.5). Equation (2.6) can be sown similarly. If M, ten A, D, and [ [ A BD A 2 AB r B D 2 = r B = r(m) = i + (M), D r[a, B = r(a), r[b, D = r(d).

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 27 Hence (2.) (2.6) reduce to (2.7). We next obtain a group of inequalities for te rank and inertia of G 1 in (1.2). Corollary 2.6. Let M and M be given by (1.1) and (1.2). Ten, r(g 1 ) max{2r[a, B r(m), r(d) 2r[B, D+r(M)}, (2.8) r(g 1 ) r(d)+2r[a, B r(m), (2.9) i ± (G 1 ) max{r[a, B i (M), i (D) r[b, D+i ± (M)}, (2.4) i ± (G 1 ) i (D)+r[A, B i (M). (2.41) Proof. Applying (1.8) and (2.25) gives [ W2 DW 2 W 1 2r(W 1 ) r W1 r(w 2 DW 2)+2r(W 1 ) = r(d)+2r(w 1 ). Substituting tese two inequalities into (2.), we obtain te first part of (2.8) and (2.9). Applying (1.9) to te first expression in (2.1), and simplifying by elementary block matrix operations, gives r(g 1 ) = r(p 1 M P 1) r[ M P 1 P 1 2r[M, P 1 +r(m) = r(d) 2r[B, D+r(M), establising te second part of (2.8). Applying (1.18), (1.2) and (2.25) gives [ W2 DW 2 W 1 r(w 1 ) i ± W1 i ± (W 2 DW 2)+r(W 1 ) = i ± (D)+r(W 1 ), i ± (G 1 ) = i ± (P 1 M P 1 ) i [ M P 1 P 1 r[m, P 1 +i ±(M) = i (D) r[b, D+i ± (M). Substituting tese inequalities into (2.6) leads to (2.4) and (2.41). Inequalities for te rank and inertia of te two submatrices G 2 and G in (1.2) can be derived similarly. Setting D = in Teorem 2.1, and simplifying, yields te following result. Corollary 2.7. Let M 1 = [ A B B, (2.42)

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela 28 Y. Tian were A C m m and B C m n, and denote its Moore Penrose inverse by [ M 1 = G1 G 2 G, (2.4) 2 G were G 1 C m m Under te condition A,, G 2 C m n and G C n n. Ten, r(g 1 ) = 2r[A, B r(m 1 ), (2.44) r(g 2 ) = r(b), (2.45) A A 2 B B r(g ) = r B A 2 B AB r(m 1 ), (2.46) B i ± (G 1 ) = r[a, B i (M 1 ), (2.47) A A 2 B B i ± (G ) = i B A 2 B AB i (M 1 ). (2.48) B r(g 1 ) = i + (G 1 ) = r[a, B r(b), (2.49) r(g ) = i (G ) = r(a)+r(b) r[a, B. (2.5) Equalities for te rank and inertia of te submatrices in (2.42), (2.4) and teir operations, suc as, A AG 1 A and A+BG B, can also be derived. Te following result was recently given in [18, Teorem.11. Teorem 2.8. Let M 1 and M 1 be given by (2.42) and (2.4). Ten, i ± (M 1 ) = i ± (A)+r(B) i ± (A AG 1 A), (2.51) i ± (M 1 ) = r(m) i (A) r(b)+i (A AG 1 A), (2.52) i ± (M 1 ) = r[a, B i (A+BG B ), (2.5) i ± (M 1 ) = r(m) r[a, B+i ± (A+BG B ), (2.54) r(m 1 ) = r(a)+2r(b) r(a AG 1 A), (2.55) r(m 1 ) = 2r[A, B r(a+bg B ). (2.56) Hence, (a) A AG 1 A i (M 1 ) = i (A)+r(B). (b) A AG 1 A i + (M 1 ) = i + (A)+r(B). (c) A = AG 1 A r(m 1 ) = r(a)+2r(b).

Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela Rank and Inertia of te Moore Penrose Inverse 29 (d) A+BG B i + (M 1 ) = r[a, B. (e) A+BG B i (M 1 ) = r[a, B. In addition to (1.2), oter types of generalized inverses of M can also be written in partitioned forms. For example, we can partition te Hermitian g-inverse of M in (1.1) as [ A B B = D [ G1 G 2 G 2 G were G 1 C m m, G 2 C m n and G C n n. Ten, te rank and inertia of G 1, G 2 and G may vary wit respect to te coice of M. In suc a case, it would be of interest to consider te maximal and minimal possible ranks and inertias of G 1, G 2 and G. In an earlier paper [1, Jonson and Lundquist defined te inertia of Hermitian operator in a Hilbert space, and gave some formulas for te inertias of Hermitian operators and teir inverses. Under tis general frame, it would be of interest to extend te results in tis paper to inertias of Hermitian operators in a Hilbert space., Acknowledgments. Te autor tanks te referee and Professor Mose Goldberg for teir elpful suggestions and comments. REFERENCES [1 A. Ben-Israel and T.N.E. Greville. Generalized Inverses: Teory and Applications. Second ed., Springer, New York, 2. [2 D.S. Bernstein. Matrix Matematics: Teory, Facts and Formulas. Second ed., Princeton University Press, Princeton, 29. [ M. Fiedler and T.L. Markam. Completing a matrix wen certain entries of its inverse are specified. Linear Algebra Appl., 74:225 27, 1986. [4 W.H. Gustafson. A note on matrix inversion. Linear Algebra Appl., 57:71 7, 1984. [5 E.V. Haynswort. Determination of te inertia of a partitioned Hermitian matrix. Linear Algebra Appl., 1:7 81, 1968. [6 E.V. Haynswort and A.M. Ostrowski. On te inertia of some classes of partitioned matrices. Linear Algebra Appl., 1:299 16, 1968. [7 L. Hogben. Handbook of Linear Algebra. Capman & Hall/CRC, 27. [8 R.A. Horn and C.R. Jonson. Matrix Analysis. Cambridge University Press, Cambridge, 1985. [9 C.H. Hung and T.L. Markam. Te Moore Penrose inverse of a partitioned matrix [ A C B D Linear Algebra Appl., 11:7 86, 1975. [1 C.R. Jonson and M. Lundquist. Operator matrix wit cordal inverse patterns. Operator Teory Adv. Appl., 59:24 251, 1992. [11 C.R. Jonson and M. Lundquist. An inertia formula for Hermitian matrices wit sparse inverses. Linear Algebra Appl., 162-164:541 556, 1992..

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