Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9236281, 14 pages
http://dx.doi.org/10.1155/2016/9236281

Research Article

Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

Xiaoji Liu,¹ Hongwei Jin,¹ and Jelena Višnjić²

¹Faculty of Science, Guangxi University for Nationalities, Nanning 530006, China
²Faculty of Medicine, University of Niš, 18000 Niš, Serbia

Received 15 July 2016; Accepted 6 September 2016

Academic Editor: Gen Qi Xu

Copyright © 2016 Xiaoji Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Representations of -inverses, -inverses, and Drazin inverse of a partitioned matrix related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which -inverses, -inverses, and group inverse of a block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

1. Introduction

Let denote the set of all complex matrices. We use , , and to denote the range, the null space, and the rank of a matrix , respectively. The smallest nonnegative integer such that is the index of which is denoted by . If , the Drazin inverse of is the unique matrix satisfying

for all . The Drazin inverse of is denoted by . In particular, when , the matrix is called the group inverse of and it is denoted by .

Recall that the Moore-Penrose inverse of a matrix is a matrix which satisfies

The Moore-Penrose inverse of is unique and it is denoted by .

For any , let denote the set of all which satisfy equations of (2). In this case is a -inverse of , which we denote by . Evidently, .

For a complex matrix of the form

in the case when is invertible, the Schur complement of in is defined by . Similarly, if is invertible, then the Schur complement of in is defined by .

It is well-known that if is nonsingular then the invertibility of a matrix is equivalent to the invertibility of the Schur complement of in and in that case the inverse of is given by

Expression (4) is called the Banachiewicz-Schur form of the matrix . The notation of Schur complement was first introduced by Schur [1]. G. E. Trapp first defined the generalized Schur complement where the ordinary inverse was replaced by the generalized inverse. For a matrix given by (3) and any fixed generalized inverse , the generalized Schur complement of in is defined by

Similarly, for some fixed , the generalized Schur complement of in is defined by

The Schur complements and generalized Schur complements were studied by a number of authors, which have applications in statistics, matrix theory, electrical network theory, discrete-time regulator problem, sophisticated techniques, and some other fields. It has been evident that the Schur complement plays an important role in many aspects of a matrix theory (see [2–8]).

Let

where , , and , and let

where , , and .

For convenience, we will firstly state the following notations which are helpful in the proofs. We denote

In this paper, we will establish necessary and sufficient conditions under which matrices and given by (7) and (8), respectively, belong to a given class of generalized inverses of the matrix given by (3). Our interest focuses on the cases when -inverses, -inverses, and group inverse of a block matrix are given by the Banachiewicz-Schur form. Also, we give representations of the Drazin inverse of block matrix under some conditions relating to the Schur complement of , which generalized the results studied by Cvetković-Ilić [9]. Also, the quotient property and the first Sylvester identity based on the MP-inverse, group inverse, and Drazin inverse are given.

We need some auxiliary lemmas.

Lemma 1 (see [10]). Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

the last two conditions being independent of the choice of and involved in and .

Lemma 2 (see [10]). Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

the last two conditions being independent of the choice of and involved in and .

Lemma 3 (see [10]). Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

Lemma 4. Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

the last two conditions being independent of the choice of and involved in and .

Proof. The conclusion follows by Lemmas 1 and 2.

Lemma 5. Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

the last two conditions being independent of the choice of and involved in and .

Using the same method as in [10], we can prove the following.

Lemma 6. Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

the last two conditions being independent of the choice of and involved in and .

Lemma 7. Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

the last two conditions being independent of the choice of and involved in and .

Lemma 8 (see [11]). Let , , and be defined by (3), (5), and (7), respectively. Then if and only if

2. Representations of -Inverse, -Inverse, and Group Inverse in terms of Banachiewicz-Schur Forms

Baksalary and Styan [10] presented the necessary and sufficient conditions under which , , , , and -inverses of block matrix can be represented by the Banachiewicz-Schur form.

Sheng and Chen [12] considered sufficient conditions under which , , and can be represented by both of the Banachiewicz-Schur forms at the same time.

Cvetković-Ilić [9] gave necessary and sufficient conditions which ensure the representation of the MP-inverse of a block matrix by both of the Banachiewicz-Schur forms.

In this section, we will present necessary and sufficient conditions under which , -inverses and the group inverse of a partitioned matrix can be represented by the Banachiewicz-Schur form.

For convenience, we first introduce the following notations:

Theorem 9. Let , , and be defined by (3), (5), and (6), respectively. Then the following happens.
(i) if and only if for arbitrary , and ,

and for any and any , there exists such that

(ii)   if and only if for arbitrary , and ,

and for any and any , there exists such that

(iii)   if and only if for arbitrary , ,

and for any and any , there exists such that , , and , and for any and , there exists such that , , and .

Proof. By Lemma 6, we have that arbitrary belongs to . Now, we only need to prove that . Take arbitrary

By assumption, for , there exists -inverse of which we denote by such that , , and . Let . We will prove that . By computation, we get

which implies . Also,

so, . Since , , and , a simple computation shows that

Thus,

Therefore, .
Since and , by Lemmas 4 and 6, we get that, for arbitrary , , and , the following hold: , , and . Moreover, by , it follows that, for any , there exists some such that . Hence, for any , , there exist and such that

A simple computation shows that

thus, we get . Since

we get ; that is, . Similarly,

which implies .
Similar to , since and , we have . Now, we will prove that . Take arbitrary

For , there exists -inverse of which we denote by such that , , and . Let . A simple computation shows that

Hence,

Therefore, . The reverse part of the proof is similar to .
Combining and , we get .

The case for -inverses is treated completely analogously and the corresponding result follows by taking adjoint.

Theorem 10. Let , , and be defined by (3), (5), and (6), respectively. Then the following happens.
(i)   if and only if for arbitrary , , and ,

and for any and any , there exists such that

(ii)   if and only if for arbitrary , , and ,

and for any and any , there exists such that

(iii)   if and only if for arbitrary , ,

and for any and any , there exists such that , , and , and for any and any , there exists such that , , and .

Similarly, we get the following results.

Theorem 11. Let , , and be defined by (3), (5), and (6), respectively. Then the following happens.
(i) For arbitrary given by (7), there exists such that if and only if and and there exist and such that

(ii) For arbitrary given by (8), there exists such that if and only if and and there exist and such that

Theorem 12. Let , , and be defined by (3), (5), and (6), respectively. Then the following happens.
(i) For arbitrary given by (7), there exists such that if and only if and and there exist and such that

(ii) For arbitrary given by (8), there exists such that if and only if and and there exist and such that

Theorem 13. Let , , and be defined by (3), (5), and (6), respectively. Then

if and only if

Proof. By Theorems 11 and 12, we get

if and only if

Since and , then . Thus,

implies

Now, we only need to prove that

is equivalent to (46). Denote . By , , and , a simple computation shows that

Similarly, by , , and , we get . Moreover, and . Thus, and and . Therefore, (51) is equivalent to (46).

Corollary 14 (see [9]). Let , , and be defined by (3), (5), and (6), respectively. Then

if and only if

Proof. By Corollary 14 from [9], it is easy to conclude that (54) is equivalent to (46).

Corollary 15. Let , , and be defined by (3), (5), and (6), respectively. Then

if and only if

Corollary 16. Let , , and be defined by (3), (5), and (6), respectively. Then

if and only if

Theorem 17. Let , , and be defined by (3), (5), and (6), respectively. Then

if and only if one of the following conditions holds.
(i)

(ii)

Proof. By Lemma 8,

if and only if

Similarly,

if and only if

Since the group inverse is unique if it exists, combining (63) and (65), we get

if and only if

Now, we only need to prove (67) is equivalent to (60). Denote . If (60) holds, then

Now, it is easy to get

and . Thus, . Hence, and . Now, we get and , which means (60) implies (67). Obviously, (67) implies (60). Thus, (67) is equivalent to (60).
The proof is similar to the proof of .

3. Representations of Drazin Inverse in terms of Banachiewicz-Schur Forms

The Drazin inverse of a square matrix has various applications in singular differential equations and singular difference equations, Markov chains, and iterative methods. Actually, in 1979 Campbell and Meyer [13] posed the problem of finding an explicit representation for the Drazin inverse of a complex block matrix , in terms of its blocks. No formula for has yet been offered without any restrictions upon the blocks. Many papers have considered this open problem and each of them offered a formula for the Drazin inverse and specific conditions for the block matrix (see [9, 11, 12, 14–22]).

Wei [23] proved that can be expressed by if , , , , and , where and .

Cvetković-Ilić [9] generalized the result from [23] and gave the representation of under simpler conditions than in [23].

Sheng and Chen [12] proved that, under the conditions , , , , and , can be represented by both of Banachiewicz-Schur forms.

In the following, we will present conditions more general than those in [9, 12, 23] which ensure the representation of by the Banachiewicz-Schur form.

Theorem 18. Let be defined by (3) and be the generalized Schur complement of in . If

then

Proof. Denote the right side of (71) by . By simple computations, we show that

Since , , , and , (72) reduce to

Obviously, . On the other hand,

Now, we need to prove that is a nilpotent matrix. Since

by , , we get

which is nilpotent and .

Corollary 19 (see [9]). Let be given by (3) and be the generalized Schur complement of in . If

and if there exists a nonnegative integer such that , then

where .

Corollary 20 (see [23]). Let be given by (3) and be the generalized Schur complement of in . If

then

Theorem 21. Let be defined as in (3) and be the generalized Schur complement of in . If

then

Proof. The proof is analogous to that of Theorem 18.

Theorem 22. Let be defined by (3) and , be the generalized Schur complement of in and in , respectively. If any of the conditions hold

(i), , , , , ,(ii), , , , , ,

then

Proof. Denote . Firstly, we prove . It is easy to see

by , and , and

by , , and . Thus, . Moreover,

by using . Therefore, . Since and , we get , . Now, we can deduce that

by , , , and and

by , , , and . Then,

follows by the uniqueness of the Drazin inverse.
Similarly, we denote . In the same way, we can easily get by , , , and by , , and . Since

by using , we obtain . Thus, we have which implies and , where we used and . Now, by , , , and , we get

and by , , , and , we have

Therefore,

Corollary 23 (see [12]). Let be defined by (3) and , be the generalized Schur complement of in and in , respectively. If

then

Proof. According to the proof of Theorem 22(ii), we have which implies by . Now, it is sufficient to conclude that (95) holds.

4. Quotient Property and the First Sylvester Identity in terms of the Generalized Schur Complement

Crabtree and Haynsworth in [18] showed a quotient formula for Schur complement of a matrix. After that, the formula was extended by Ostrowski in 1971 [24] and in 1973 [19].

Let

If and are square and nonsingular, then

This is the so-called quotient property which is proven by Crabtree and Haynsworth in [18].

The first Sylvester identity is that

where and are nonsingular.

Sheng and Chen [12] studied the quotient properties and the first Sylvester identity with respect to MP, group, and Drazin inverses. Cvetković-Ilić in [9] derived the properties under conditions different than those used in [12].

In this section, we will give the quotient properties and the first Sylvester identity of MP, group, and Drazin inverses under the conditions which are much simpler than those of [9, 12]. Notice that the proofs of the following results are almost identical to the proofs of the corresponding results from [12]. We will state the theorems without proofs.

We first give the quotient properties based on MP, group, and Drazin inverses. For convenience, we denote

Theorem 24. Let and . If satisfies the conditions of Corollary 15, then

Theorem 25. Let and . If satisfies the conditions of Theorem 11(i) or (ii), then

Theorem 26. Let and . If satisfies the conditions of Theorem 18, then

The first Sylvester identity with respect to MP, group, and Drazin inverses is expressed as follows.

Theorem 27. Let , , , , and be the forms of (97). If satisfies the conditions of Corollary 15, then

Theorem 28. Let , , , , and be the forms of (97). If satisfies the conditions of Theorem 11(i) or (ii), then

Theorem 29. Let , , , , and be the forms of (97). If satisfies the conditions of Theorem 18, then

5. Applications of the Generalized Schur Complement

The Schur complements and generalized Schur complements were shown playing an important role in various applied mathematical settings, such as statistics, matrix theory, electrical network theory, discrete-time regulator problem, sophisticated techniques, and many other fields.

In this section, we will give an application of the generalized Schur complement in the solution of a linear system. Using the generalized Schur complement, we can split a larger system into two small linear systems by the following steps.

Let

be a linear system. Applying the block Gaussian elimination to the system, we have

Hence, we get

that is,

Now, the solution of system (107) can be obtained by the two small linear systems above. In this case, the operation can be significantly simplified. We will also notice that the Moore-Penrose inverse of can be replaced by other generalized inverses, such as the group inverse, the Drazin inverse, and the generalized inverse of or even the ordinary inverse .

In the following, we will give the group inverse solutions of the linear system.

Theorem 30. Let

be a linear system. Suppose satisfies all the conditions of Theorem 17, partitioning and as

which have appropriate sizes with . If , then the solution of linear system (111) can be expressed as

where .

Proof. Since , we conclude that is the solution of linear system (111). By Theorem 17, we can get the following:

Now, it is easy to see the solution can be expressed as

which is also the group inverse solutions of the two small linear systems of (110), respectively.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11361009). The third author is supported by Grant no. 174007 of the Ministry of Education, Science and Technological Development, Serbia.

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