Generalized Schur complements of matrices and compound matrices

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1 Electronic Journal of Linear Algebra Volume 2 Volume 2 (200 Article Generalized Schur complements of matrices and compound matrices Jianzhou Liu Rong Huang Follow this and additional wors at: Recommended Citation Liu, Jianzhou and Huang, Rong. (200, "Generalized Schur complements of matrices and compound matrices", Electronic Journal of Linear Algebra, Volume 2. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

2 GENERALIZED SCHUR COMPLEMENTS OF MATRICES AND COMPOUND MATRICES JIANZHOU LIU AND RONG HUANG Abstract. In this paper, we obtain some formulas for compound matrices of generalized Schur complements of matrices. Further, we give some Löwner partial orders for compound matrices of Schur complements of positive semidefinite Hermitian matrices, and obtain some estimates for eigenvalues of Schur complements of sums of positive semidefinite Hermitian matrices. Key words. Löwner partial order, generalized Schur complement, compound matrix, eigenvalues. AMS subject classifications. 5A45, 5A57.. Introduction. Many important results for compound matrices and Schur complements of matrices have been obtained in [2,3,9,0]. Recently, Smith [], Liu et al. [5,6], Li et al. [4] obtained some estimates of eigenvalues and singular values for Schur complements of matrices. Liu and Wang [7], Wang and Zhang et al. [2] obtained some Löwner partial orders of Schur complements for positive semidefinite Hermitian matrices respectively. Wang and Zhang [3] obtained some Löwner partial orders for Hadamard products of Schur complements of positive definite Hermitian matrices. Liu[8] obtained some Löwner partial orders for Kronecer products of Schur complements of matrices. In this paper, we study Schur complements of matrices and compound matrices. Let C, R and R + denote the set of complex, real, and positive real numbers respectively. Let C m n denote the set of m n complex matrices. Let N n denote the set of n n normal matrices. Let H n denote the set of n n Hermitian matrices, and let H n (H > n denote the subset consisting of positive semidefinite (positive definite Received by the editors on June 0, Accepted for publication on July 3, 200. Handling Editors: Roger A. Horn and Fuzhen Zhang. Department of Mathematical Science and Information Technology, Hanshan Normal University, Chaozhou, Guangdong 5204, China; and Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 405, P. R. China. This wor was supported in part by the National Natural Science Foundation of China 09776, the Guangdong Provincial Natural Science Foundation of China and the ey project of Hunan Provincial Natural Science Foundation of China 0JJ liujz@xtu.edu.cn Department ofmathematics and Computational Science, Xiangtan University, Xiangtan, Hunan. This wor was supported by China Postdoctoral Science Foundation (No and the Research Foundation of Education Bureau of Hunan Province (No. 09C942. 2

3 Generalized Schur complements of matrices and compound matrices 3 Hermitian matrices. For A C m n, the ran of A is denoted by r(a. Denote by A the conjugate transpose matrix of A. For A,B H n, write B A if B A H n. The relation is calledthe Löwnerpartial order. For A C n n, we alwaysarrange the eigenvalues of A as λ (A λ n (A. For A C m n, denote the column space of A by R(A. Let be an integer with n. Define (. Q,n = {ω = {ω,...,ω } : ω i R and ω < < ω n}. Given a matrix A = (a ij C m n. Let and r be integers satisfying m and r n, respectively. If α Q,m and β Q r,n, then A(α,β denotes the r matrix whose (i,j entry is a αi,β j. If α is equal to β, A(α α is abbreviated to A(α. Let A C m n, l = min{m,n}, L = {,2,...,l}. We denote by C (A the th compound matrix. Let all the elements of Q,m be ordered lexicographically; { ( } m denotes the lexicographical order. Let Q r,m = α i i =,..., satisfy α α 2 α ( m r r. Define a mapping σ : σ(α i = i; it is a one to one correspondence. We denote σ(α by j α if α Q r,m. If α Q r,m and β Q,n, then A jα,j β denotes the (j α,j β entry of C (A. Let A C m n. If X C n m satisfies the equations (.2 (i AXA = A, (ii XAX = X, (iii (XA = XA, (iv (AX = AX, then X is called the Moore-Penrose (MP inverse of A. Let A C m n, α M, β N, α = M α, and β = N β. Then (.3 A/ + (α,β = A(α,β A(α,β[A(α,β] + A(α,β is called the generalized Schur complement with respect to A(α,β. If A(α,β is a nonsingular matrix, then A/ + (α,β = A/(α,β is called the Schur complement with respect to A(α,β. If α = β, we define A/ + (α,β = A/ + α and A/(α,β = A/α respectively. In [], Ando shows that if A,B H n, then and (A+B/α A/α+B/α, A 2 /α (A/α 2. In this paper, we provide some similar results for compound matrices of the Schur complements of positive semidefinite Hermitian matrices and obtain some estimates for eigenvalues.

4 4 Jianzhou Liu Rong Huang 2. Some formulae for compound matrices of generalized Schur complements of matrices. In this section, using properties of compound matrices and M P inverses, we obtain some formulae for compound matrices of generalized Schur complements of matrices. Lemma 2.. Let A C m n. Then (2. C (A + = [(C (A] +. Proof. By properties of compound matrices, we have i. C (A = C (AA + A = C (AC (A + C (A, ii. C (A + = C (A + AA + = C (A + C (AC (A +, iii. C (A + A = C (A + C (A, iv. C (AA + = C (AC (A +. Thus, by equations (i-(iv of (.2, we easily get that C (A + = [C (A] +. (2.2 Lemma 2.2. ([4] Let A C m n be partitioned as ( A A A = 2 A 2 A 22, where and ( A2 r A 22 ( A r A 2 = r(a,a 2 = r(a, = r(a 2,A 22 = r(a 22 A 2 A + A 2. Then (2.3 ( A A + + = +A + A 2S + A 2 A + A + A 2S + S + A 2 A + S +, where S = A 22 A 2 A + A 2. Lemma 2.3. Let A Hn be partitioned as (2.2 with r(a = r(a +r(a 22. Then (2.3 holds.

5 Generalized Schur complements of matrices and compound matrices 5 Theorem 2.4. Let A C m n, α M, β N, α = M α and β = N β. Set min{ M α, N β }. Suppose that the following conditions are satisfied: ( A(α,β (2.4 r = r(a(α,β,a(α,β = r(a(α,β, A(α,β (2.5 ( A(α,β r A(α, β = r(a(α,β,a(α,β = r(a/ + (α,β; and (2.6 r ( C [A(α,β ] C (A(γ,δ = r(c [A(α,β ],C (A(γ,δ = r{c [A(α,β ]}, (2.7 ( C (A(γ,δ r C (A(γ,δ = r(c (A(γ,δ,C (A(γ,δ = r(c (A/ + (γ,δ, where γ = {j ᾱ α, = Q ᾱ ᾱ,ᾱ,m }, δ = {j β β β, β =, β ( ( m Q,n }; and γ = {,2,..., } γ n, δ = {,2,..., } δ. Then (2.8 C [A/ + (α,β] = C (A/ + (γ,δ. Proof. For A C m n, there exist permutation matrices P C m m and Q C n n such that ( A(α,β A(α,β PAQ =. A(α,β A(α,β Let α = {,2,..., α }, β = {,2,..., β }, α = M α, β = N β, γ = {,2,..., γ }, δ = {,2,..., δ }, ( m γ = {,2,..., ( } γ n, δ = {,2,..., } δ.

6 6 Jianzhou Liu Rong Huang Thus, by (2.3, (2.4 and Lemma 2.2, we have (2.9 By (2.5, (2.6 and Lemma 2.2, we have (2.0 [A + (α,β ] + = A/ + (α,β. {[C (A] + (γ,δ } + = C (A/ + (γ,δ. Therefore, from (.3, (2.8 and (2.9, it follows that C [A/ + (α,β] = C [(PAQ/ + ( α, β] = C {[(PAQ + ( α, β ] + } (by (2.8 = {C [(PAQ + ( α, β ]} + (by (.3 = {C [(PAQ + ]( γ, δ } + = {[C (PAQ] + ( γ, δ } + (by (.3 = C (PAQ/ + ( γ, δ (by (2.9 = C (A/ + (γ,δ. Corollary 2.5. Let A C n n, α N and α = N α. Set n α. If A, A(α, and A(α are nonsingular respectively, then (2. C (A/α = C (A/γ where γ = {j ᾱ ᾱ α, ᾱ =,ᾱ Q,n } and γ = {,2,..., ( n } γ. In a manner similar to the proof of Theorem 2.4, we obtain the following result by using Lemma 2.3. Theorem 2.6. Let A H n, α N and α = N α. Set n α. Suppose that the following conditions are satisfied: r(a = r(a(α +r(a(α, r[c (A] = r[c (A(γ]+r[C (A(γ ], ( where γ = {j ᾱ α n ᾱ, ᾱ =,ᾱ Q,n }, γ = {,2,..., (2.2 C (A/ + α = C (A/ + γ. } γ. Then Corollary 2.7. Let A H > n, α N and α = N α. Set n α. Then (2.3 C (A/α = C (A/γ, where γ = {j ᾱ ᾱ α, ᾱ =,ᾱ Q,n }, γ = {,2,..., ( n } γ.

7 Generalized Schur complements of matrices and compound matrices 7 3. Some Löwner partial orders for compound matrices of sums of matrices. In this section, we obtain some Löwner partial orders for compound matrices of Schur complements of positive semidefinite Hermitian matrices. Further, we obtain some estimates for eigenvalues of Schur complements of sums of positive semidefinite Hermitian matrices. Lemma 3.. ([2, p. 84] Let A,B H n. Then (3. λ t (A+B max i+j=n+t {λ i(a+λ j (B}. Lemma 3.2. (i Let A H n >, N and r R. Then (3.2 C (A r = [C (A] r. (ii Let A Hn, N and r R +. Then (3.2 holds. Proof. Since A H n >, there exists an unitary matrix U such that A = Udiag(λ (A,...,λ n (AU where λ i (A > 0 (i =,2,...,n. Thus C (A r = C [Udiag(λ r (A,...,λ r n(au ] = C (UC [diag(λ r (A,...,λr n (A][C (U] = C (Udiag([λ (A...λ (A] r,...,[λ n + (A...λ n (A] r [C (U] = {C (Udiag(λ (A...λ (A,...,λ n + (A...λ n (A[C (U] } r = {C (UC [diag(λ (A,...,λ n (A][C (U] } r = {C [Udiag(λ (A,...,λ n (AU ]} r = [C (A] r. In a manner similar to the proof of (i, we obtain (ii. Lemma 3.3. Let A,B Hn, N. Then (3.3 C (A+B C (A+C (B. Proof. Since A,B H n, we have (3.4 C (A+B = C (A 2 A 2 +B 2 B 2 = C [(A 2,B 2 (A 2,B 2 ] = C [(A 2,B 2 ]{C [(A 2,B 2 ]}.

8 8 Jianzhou Liu Rong Huang It is not difficult to show that there exist X and a permutation matrix U such that (3.5 where X is ( n C [(A 2,B 2 ] = (C (A 2,C (B 2,XU, ] [( 2n ( n 2 Therefore, by (3.3 and (3.4, we have and U is ( 2n ( 2n C (A+B = [(C (A 2,C (B 2,XU][C (A 2,C (B 2,XU] = ([C (A] 2,[C (B] 2,X([C (A] 2,[C (B] 2,X = C (A+C (B+XX C (A+C (B.. Lemma 3.4. Let A,B H n,a B and N. Then (3.6 C (A C (B. Proof. Lemma 3.3 ensures that C (A = C [B +(A B] C (B+C (A B C (B. Theorem 3.5. Let A,B H n, α N, and α = N α. Set n α. Suppose that the following conditions are satisfied: (3.7 r(a = r(a(α] +r[a(α ], r(b = r[b(α]+r[b(α ], r[c (A] = r[c (A(γ]+r[C (A(γ ], r[c (B] = r[c (B(γ]+r[C (B(γ ], (3.8 ( where γ = {j ᾱ α n = Q ᾱ, ᾱ,ᾱ,n } and γ = {,2,..., } γ. Then (3.9 C [(A+B/ + α] C (A/ + γ +C (B/ + γ. Proof. By [8, Theorem 3.], it follows that (3.0 (A+B/ + α A/ + α+b/ + α. Thus, by (3.5, (3.9, (3.2, (3.6, (3.7 and (2.3, we conclude that C [(A+B/ + α] C (A/ + α+b/ + α (by (3.5 and (3.9 C (A/ + α+c (B/ + α (by (3.2 = C (A/ + γ +C (B/ + γ. (by (3.6,(3.7 and (2.3.

9 Generalized Schur complements of matrices and compound matrices 9 Corollary 3.6. Let all assumptions of Theorem3.5 be satisfied. If A B H n, then (3. C [(A B/ + α] C (A/ + γ C (B/ + γ. Proof. Since A B Hn, Theorem 3.5 ensures that C (A/ + γ = C (A/ + α = C [(B +(A B/ + α] is at least C (B/ + α+c [(A B/ + α] = C (B/ + γ +C [(A B/ + α], which means that (3. holds. Theorem 3.7. Let all the assumptions of Theorem 3.5 be satisfied. Then (3.2 λ t [(A+B/ + α] is bounded below by the maximum of λ t (A/ + α+ λ n α t+ (B/ + α and λ n α t+ (A/ + α+ λ t (B/ + α. Proof. Theorem 3.5 and (2.3 imply that λ t [(A+B/ + α]λ {C [(A+B/ + α]} λ [C (A/ + α+c (B/ + α, which is bounded below by the maximum of λ [(C (A/ + α]+λ ( n α [C (B/ + α] and λ ( n α [C (A/ + α+λ [(C (B/ + α],

10 20 Jianzhou Liu Rong Huang and hence by the maximum of and Remar 3.8. following result λ t (A/ + α+ λ n α t+ (B/ + α λ n α t+ (A/ + α+ λ t (B/ + α. In a manner similar to the proof of Theorem 3.7, we get the (3.3 λ n α t+ [(A+B/ + α] λ n α t+ (A/ + α+ λ n α t+ (B/ + α. Theorem 3.9. Let B H n, L = {,2,...,min{m,n}}, α L, and α = M α, β = N α. Set L α. If A C m n satisfies conditions (2.3-(2.6 and (3.4 R[A(α,α ] R[A(α], then (3.5 C [(ABA / + α] [C (A/ + γ]c (B(δ [C (A/ + γ], where γ = {j ᾱ α = Q ᾱ, ᾱ,ᾱ,m }, δ = {j β β β, β =, β Q,m }, ( ( m and γ = {,2,..., } γ n, δ = {,2,..., } δ. Proof. Using (3.3, in a manner similar to the proof of [0, Theorem 3] and [9, Theorem 2 ], it follows that (3.6 (ABA / + α (A/ + αb(α (A/ + α. Thus, from (3.5, (3.5 and (2.7, we obtain C [(ABA / + α] C [(A/ + αb(β (A/ + α ] (by (3.5 and (3.5 = C (A/ + αc [B(β ]C [(A/ + α ] = [C (A/ + γ]c (B(δ [C (A/ + γ] (by (2.7. Theorem 3.0. Let all assumptions of Corollary 2.7 be satisfied, and 0 l. Then (3.7 C (A l /α [C (A/γ] l.

11 Generalized Schur complements of matrices and compound matrices 2 Proof. By [3], for t +, we have (3.8 A(α [A t (α ] t. Replace A with (A t in Eqs. (3.7, and let l = t. Then (3.9 (A l (α = (A l (α [A (α ] l. It is nown by [3, p. 474] that for B H > n, (3.20 B (α = (B/α. Thus, by (3.8 and (3.9, we get (3.2 A l /α (A/α l. Therefore, from (3.20, (3.5, (3., and (2.2, we have C (A l /α C [(A/α l ] = [C (A/α] l = [C (A/γ] l. 4. Some Löwner partial orders for compound matrices of Schur complements of two types matrices. Let A C n n. Then (4. H A = A+A 2, S A = A A. 2 In this section, we study compound matrices of Schur complements of complex square matrices that are either normal or have positive definite Hermitian part. Theorem 4.. Let all assumptions of Corollary2.5 be satisfied. If (A+A (α H > α, then (4.2 [ ] 2 deta C (A+A /γ C [A/α+(A/α ] C (A+A /γ. deth A Proof. By [9, Theorem 7 and Theorem 8], we have (4.3 [ ] 2 detha [A/α+(A/α ] (A+A /α A/α+(A/α. deta Thus (4.4 [ ] 2 deta (A+A /α A/α+(A/α (A+A /α. deth A

12 22 Jianzhou Liu Rong Huang By (4.3, (3.5 and (2.0, we get that { [ deta ] 2 C [A/α+(A/α ] C (A+A /α} deth A (by(4.3 and (3.5 [ ] 2 deta = C [(A+A /α] deth A [ ] 2 deta = C (A+A /γ (by (2.0, deth A and C [A/α+(A/α ] C [(A+A /α] = C (A+A /γ. Theorem 4.2. Let A N n, α N and α = N α. Set n α. Suppose that the following conditions are satisfied: r(h 2 A = r[h2 A (α]+r[h2 A (α ], r(s 2 A = r[s 2 A(α]+r[S 2 A(α ], r[c (H 2 A ] = r[c (H 2 A (γ]+r[c (H 2 A (γ ], r[c (S 2 A] = r[c (S 2 A(γ]+r[C (S 2 A(γ ], ( where γ = {j ᾱ α n = Q ᾱ, ᾱ,ᾱ,n }, γ = {,2,..., } γ. Then (4.5 C [(AA / + α] [C (H A ] 2 / + γ +( [C (S A ] 2 / + γ. Proof. Since A N n, we have (4.6 H A S A = S A H A. Noting that S A = S A, by (4.6, we obtain AA = (H A +S A (H A +S A = (H A +S A (H A +S A (4.7 = H 2 A +H A S A +S A H A +S A S A = H 2 A +S A S A. By (4.6, (3.9, (3.5, (3.2 and (2., we have C [(AA / + α] = C [(H 2 A +S A S A/ + α] (by (4.6

13 Generalized Schur complements of matrices and compound matrices 23 C [HA 2 / +α+(s A SA / +α] (by (3.9 and (3.5 C (HA 2 / +α+c [(S A SA / +α] (by (3.2 = C (HA 2 / +γ +C (S A SA / +γ (by (2. = [C (H A ] 2 / + γ +C ( SA/ 2 + γ = [C (H A ] 2 / + γ +( [C (S A ] 2 / + γ. Theorem 4.3. Let all the assumptions of Corollary 2.5 be satisfied. Suppose A N n is nonsingular and each of H A and S A is nonsingular. Then (4.8 C [(AA /α] [C (H A S A H A S A] 2 /γ +( [C (S A H A S A H A] 2 /γ. Proof. Since A N n, we have A N n. Further Similarly, we have H A = 2 (A +(A = 2 A (A+A (A = (A H A A = [(H A +S A H A (H A +S A ] = (H A S A H A S A. S A = 2 (A A = (S A H A S A H A. Thus, in a manner similar to the proof of Theorem 4.2, we obtain ( Conclusions. We have obtained some formulae for compound matrices of generalized Schur complements of matrices. Using these results, we studied some Löwner partial orders for compound matrices of Schur complements of positive semidefinite Hermitian matrices. If A,B Hn, we extend some results in [] and show that C [(A+B/ + α] C (A/ + γ +C (B/ + under some restrictive conditions, as shown in Theorem 3.5, as well as C (A l /α [C (A/γ] l if A H n >, as shown in Theorem 3.0. In addition, we provide some results for compound matrices of Schur complements of complex square matrices that are either

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