GUO Mei-hua，LIU Ding-you

(School of Mathematics and Statistics，Wuhan University，Wuhan 430000，China)



Article ID：1000-5641(2016)04-0038-06

Expressions on generalized inverses of the Schur complement of a 2×2 block matrix

GUO Mei-hua，LIU Ding-you

(School of Mathematics and Statistics，Wuhan University，Wuhan 430000，China)

2×2 block matrix；generalized inverse；Schur complement

0　 Introduction

In the paper，let Cm×nrepresent the set of all m×n complex matrices，and A∗represent the conjugate transpose of A. A generalized inverse X∈Cn×mof A∈Cm×nis a matrix which satisfies the following one or more equations

(1)AXA = A；

收稿日期：2015-06
基金项目：国家自然科学基金(11371284)
第一作者：郭美华，女，硕士生，研究方向为矩阵分析及其应用. E-mail：guomeihuaxq@whu.edu.cn.通信作者：刘丁酉，男，教授，研究方向为矩阵分析及其应用. E-mail：liudingyou487@163.com.

(2)XAX = X；

(3)(AX)∗= AX；

(4)(XA)∗= XA.

Let{i，j，k}be a subset of{1，2，3，4}. The matrix which satisfies equations(i)，(j)and(k)will be called an{i，j，k}-inverse of A，denoted by A(i，j，k)，while the collection of all A(i，j，k)is denoted byA{i，j，k}. Specially，we denote the matrix which satisfies the equation(1)by A-. The collection of all A-is denoted by A{1}. The generalized inverse X is unique if it satisfies all the four equations，which is called the Moore-Penrose inverse of A，denoted by A+(cf.[1]). Some detailed properties can refer to the paper [2]. Moreover，we use PA= Im- AA-and QA= In- A-A.

The motivation for this paper is the article of Yong-ge Tian and Y. Takane[3]，in which they derived various necessary and sufficient conditions on Moore-Penroseinverse with Banachiewicz-Schur forms and inspired us to consider the relationship between two Schur complement matrices.

We consider a matrix M∈C(m+n)×(p+q)，partitioned as

where A∈Cm×pand D∈Cn×q. As we all know，if A and D in(1)are all square and nonsingular，M can be decomposed as

Based on the previous conditions，if M∈C(m+q)×(m+q)is square and nonsingular as well，the Schur complement Z∈Cq×qof A and the Schur complement S∈Cm×mof D in M，defined as

Z = D - CA-1B，S = A - BD-1C，

are also nonsingular. Then the inverse of M can be written as the two forms

Thus，we get S-1= A-1+ A-1BZ-1CA-1.

When the matrices M，A，D are singular，the relation between Z = D-CA-B and S = A - BD-C is various. In 1994，R. Kala and K. Radoslaw derived the result of the generalized inverses of the sum of matrices under the following conditions[4]：

(1)BQD= 0，PDC = 0，BQZ= 0，PZC = 0；

(2)PAB = 0，BQD= 0，PDC = 0，BQZ= 0，PZC = 0；

(3)CQA= 0，BQD= 0，PDC = 0，BQZ= 0，PZC = 0.

In this article，we derive various expressions on the generalized inverses of the Schur complement of a 2×2 block matrix M under different conditions. For more interesting results concerning the Schur complement，please see [5-7].

1　 Main results

Theorem 1.1 Assume that PAB = 0，BQDZ-C = 0，BD-PZC = 0 and X = A-+A-BZ-CA-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(

3)X∈S{1，3}，if A-∈A{1，3}.

Proof According to the given conditions，we have

SX =(A - BD-C)(A-+ A-BZ-CA-)

= AA-+ AA-BZ-CA-- BD-CA-- BD-CA-BZ-CA-

= AA-+ AA-BZ-CA-- BD-CA-- BD-(D - Z)Z-CA-

= AA-+ AA-BZ-CA-- BD-CA-- BD-DZ-CA-+ BD-ZZ-CA-

= AA-+ AA-BZ-CA-- BZ-CA-+ BZ-CA-- BD-DZ-CA-- BD-PZCA-

= AA-- PABZ-CA-+ BQDZ-CA-- BD-PZCA-

= AA-.

It is easy to get

SXS = AA-(A - BD-C)

= AA-A - AA-BD-C

= A - BD-C

= S.

The result(2)follows by S which right multiplies SX.

The result(3)follows immediately.

In the same way，we can get a new theorem.

Theorem 1.2 Assume that CQA= 0，BQZD-C = 0，BZ-PDC = 0 and X = A-+A-BZ-CA-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，4}，if A-∈A{1，4}.

Proof We calculate XS instead of SX.

XS =(A-+ A-BZ-CA-)(A - BD-C)

= A-A - A-BD-C + A-BZ-CA-A - A-BZ-CA-BD-C = A-A - A-BD-C + A-BZ-CA-A - A-BZ-(D - Z)D-C

= A-A - A-BD-C + A-BZ-CA-A - A-BZ-DD-C + A-BZ-ZD-C

= A-A - A-BD-C + A-BZ-ZD-C + A-BZ-CA-A - A-BZ-C + A-BZ-PDC

= A-A - A-BQZD-C - A-BZ-CQA+ A-BZ-PDC

= A-A.

It is easy to obtain

XSX = A-A(A-+ A-BZ-CA-)

= A-AA-+ A-AA-BZ-CA-

= A-+ A-BZ-CA-

= X.

The result(2)follows by XS which left multiplies X. Others are obtained in the same way.

If we combine the conditions of Theorem 1.1 and Theorem 1.2，we can derive the following results.

Corollary 1.3 Assume that PAB = 0，CQA= 0，BQDZ-C = 0，BD-PZC = 0，BQZD-C = 0，BZ-PDC = 0 and X = A-+ A-BZ-CA-. If A-= A+，we have X = S+.

If we strengthen the conditions of Theorem 1.1 and Theorem 1.2，Corollary 1 of [4]is got.

Corollary 1.4 Assume that BQD= 0，PDC = 0，BQZ= 0，PZC = 0，and X = A-+ A-BZ-CA-，then

(1)X∈S{1}，if either PAB = 0 or CQA= 0，or both holds；

(2)X∈S{1，2}，if A-∈A{1，2}and either PAB = 0 or CQA= 0，or both holds；

(3)X∈S{1，3}，if A-∈A{1，3}and PAB = 0 holds；

(4)X∈S{1，4}，if A-∈A{1，4}and CQA= 0 holds；

(5)X = S+，if A-= A+and both PAB = 0 and CQA= 0 hold.

In particular，in case D = I，we have Z = I-CA-B，S = A-BC. The generalized inverses of Schur complement can be considered as the generalized inverses of subtraction between matrices.

Corollary 1.5 Assume that PAB = 0，BPZC = 0，and X = A-+ A-BZ-CA-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，3}，if A-∈A{1，3}.

Corollary 1.6 Assume that CQA= 0，BQZC = 0，and X = A-+ A-BZ-CA-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，4}，if A-∈A{1，4}.

In the same way，if we combine the assumptions of Corollary 1.5 and Corollary 1.6，we obtain Corollary 1.7.

Corollary 1.7 Assume that PAB = 0，CQA= 0，BPZC = 0，BQZC = 0，A-= A+，then X = A-+ A-BZ-CA-= S+.

On the assumption of D = I，if we presume the generalized Schur complement Z is nonsingular，it has a succinct result.

Corollary 1.8 Assume that PAB = 0，Z is nonsingular，and X = A-+A-BZ-1CA-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，3}，if A-∈A{1，3}.

Proof Since Z is nonsingular，we have PZ= 0. Therefore Theorem 1.1 goes for it.

Analogously，a new result for X appears.

Corollary 1.9 Assume that CQA= 0，Z is nonsingular，and X = A-+A-BZ-1CA-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，4}，if A-∈A{1，4}.

Corollary 1.10 Assume that PAB = 0，CQA= 0，Z is nonsingular and A-= A+，then X = A-+ A-BZ-1CA-= S+.

Analogous to Theorem 1.1，we have a result.

Theorem 1.11 Assume that PAB = 0，BQDZ-B∗= 0，BD-PZB∗= 0 and X = A-+A-BZ-B∗A-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，3}，if A-∈A{1，3}.

Since A is positive semidefinite，the condition PAB = 0 is equivalent to B∗QA= 0.

Theorem 1.12 Assume that A is positive semidefinite，PAB = 0，BQZD-B∗= 0，BZ-PDB∗= 0 and X = A-+ A-BZ-B∗A-，then

(1)X∈S{1}；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，4}，if A-∈A{1，4}.

Corollary 1.13 Assume that A is positive semidefinite，PAB = 0，BQD= 0 and X = A-+ A-BZ-B∗A-. Then

(1)X∈S{1}and X is positive semidefinite；

(2)X∈S{1，2}，if A-∈A{1，2}；

(3)X∈S{1，3}，if A-∈A{1，3}；

(4)X∈S{1，4}，if A-∈A{1，4}；

(5)X = S+，if A-= A+.

Moreover，when D = I and S is nonnegative，the case has been discussed in the paper [8]by a geometric approach introducing orthogonal projections and [9]applied in the solution of systems of linear equations obtained by using Lagrange multiplies to find a constrained minimum.

[References]

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[2]郭美华，刘丁酉.分块2次幂零矩阵的广义Schur补[J].武汉大学学报(理学版)，2015，61(6)：563-567.

[3]TIAN Y G，TAKANE Y. More on generalized inverses of partitioned matrices with Banachiewicz-Schur forms [J]. Linear Algebra and its Applications，2009，430(5/6)：1641-1655.

[4]KALA R，KLACZYNSKI K. Generalized inverses of a sum of matrices[J]. Sankhyā：The Indian Journal of Statistics，Series A，1994，56：458-464.

[5]ZHANG F Z. The Schur Complement and Its Applications[M]. New York：Springer-Verlag New York Inc，2005.

[6]OUELLETTE D V. Schur complements and statistics[J]. Linear Algebra and its Applications，1981，36：187-295.

[7]ANDO T. Generalized Schur complements[J]. Linear Algebra and its Applications，1979，27：173-186.

[8]MINAMIDE N. An extension of the matrix inversion lemma[J]. SIAM Journal on Algebraic and Discrete Methods，1985，6(3)：371-377.

[9]PRINGLE R M，RAYNER A A. Expressions for generalized inverses of a bordered matrix with application to the theory of constrained linear models[J]. SIAM Review，1970，12(1)：107-115.

(责任编辑：林磊)

10.3969/j.issn.1000-5641.2016.04.005 !

2×2分块矩阵中Schur补的广义逆表示

郭美华，刘丁酉

(武汉大学数学与统计学院，武汉430000)

2×2分块矩阵；广义逆；Schur补

O151 Document code：A