Journal of Applied Mathematics

Volume 2013 (2013), Article ID 479260, 7 pages

http://dx.doi.org/10.1155/2013/479260

Research Article

The Explicit Expression of the Drazin Inverse and Its Application

Xiaoji Liu,¹ Shaowu Huang,¹ Liang Xu,¹ and Yaoming Yu²

¹School of Science, Guangxi University for Nationalities, Nanning 530006, China

²School of Mathematical Sciences, Monash University, VIC 3800, Australia

Received 18 July 2013; Accepted 6 August 2013

Academic Editor: Renat Zhdanov

Copyright © 2013 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.

We give explicit expressions of of two matrices and , in terms of , , , and , , under the condition that , and apply the result to finding an explicit representation for the Drazin inverse of some block matrix.

1. Introduction

In recent years, the representations and perturbation analysis of the Drazin inverse for matrices or operators have been investigated (see [1–6]). In [7], the author presented the presentations of the Drazin inverse of sum and product of two operators over Banach spaces on the condition of the commutativity up to a factor. And, in [8], the same author discussed explicit representations of Drazin inverses of sums and differences of two idempotents over Hilbert spaces.

These investigations motivate us to deal with an explicit expression of the Drazin inverse of differences and sums of two matrices. The paper is organized as follows. In this section, we will introduce some notions and lemmas. In Section 2, we will present these explicit expressions of differences and sums of two matrices and under the condition . In Section 3, we will deduce an explicit representation for the Drazin inverse of the block matrix with and in terms of its subblocks and their Drazin inverses and . In Section 4, we will present a numerical example to demonstrate the main result in Section 2.

Throughout this paper the symbol stands for the set of complex matrices, and stands for the unit matrix. Let ; the Drazin inverse, denoted by , of matrix is defined as the unique matrix satisfying

where is the index of . In particular, if , then is called the group inverse, denoted by , of . Apparently, if is nonsingular, then ; otherwise, , especially . If is nilpotent, then . If is nonsingular and , then (see [6, 9, 10]). For convenience, we write and use the convention , if .

Before we start the discussion, we need some preparations.

Lemma 1 (see [11, Theorem 3.2]). Let with and . Then,

where

Lemma 2. Let with . If , then

Proof. Since and ,

and then . So from this, by induction, it follows that for .

Now, we will show inductively that . It holds for . Assume it holds for ; that is, . Then,

So it holds for any .

From , we can similarly show that .

Lemma 3. Let be nilpotent of index and . If , then

where , .

Proof. Since , we can easily write as

We will prove the relationship

by induction on . Obviously, (9) holds for . Assume inductively that (9) holds for .

Since and ,

On the other hand,

So, (9) holds for . Hence, (9) holds for .

Lemma 4 (see [9, Lemma ]). Let

Then, .

2. The Drazin Inverse of Differences and Sums of Two Matrices

In this section, we will investigate how to express as a function of , , , and , , under the condition . We begin with the following theorem, in which is assumed to be nilpotent.

Theorem 5. Let with . If and is nilpotent of index , then

where and , .

Proof. If , then, from the convention in Section 1, (13) clearly holds. Now assume . From , there exists a nonsingular matrix such that

where is nonsingular and is nilpotent of index . Partitioning conformably with , we have

Since , by Lemma 2, and then , , and , . Let be the index of nilpotent matrix . Then,

and therefore is also nilpotent, and . Thus,

By Lemma 1, we have

where

By Lemma 3,

Thus,

So, by (18),

If , then . So, it immediately follows from the above theorem.

Corollary 6. Let with . If and is nilpotent of index , then

where , , .

If the nilpotency of is taken out in Theorem 5, then we can obtain our main result, a more general result.

Theorem 7. Let with , , , and . If , then

where , , .

Proof. There exists a nonsingular matrix such that

where is nonsingular and is nilpotent of index . From , we get

where and . So

By Lemma 1, we have

where

and and .

Since and , by Theorem 5 and Lemma 2,

and then

Thus,

Since

we have that, for ,

Obviously,

where the symbol denotes or . Also,

Note that . Then, by (32),

Note that . So, by Theorem 5,

Hence, putting (34), (37), and (38) into (28) yields (24).

Corollary 8. Let with , , , and . If , then

where , , .

3. The Drazin Inverse of Some 2 × 2 Block Matrix

In this section, we will apply the results in Section 2 to studying the representation for the Drazin inverse of a block matrix, in terms of its subblocks.

Theorem 9. Let with , , , and . If and , then

where , , .

Proof. Let

Then, , and, for ,

where the subscript stands for or its absence.

Since , where denotes or . Thus,

So, for ,

Also, for ,

and, for ,

Since

we have

and then, by Lemma 4, , and .

By Lemma 4, . Further, by (43), .

Hence, putting (45)~(49) into (39) yields (40).

4. Example

In this section, we present a numerical example to demonstrate Theorem 7.

Example 1. Taking as follows:

we can get

and , , , and . By Theorem 7, we have

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