Journal of Applied Mathematics

Volume 2013, Article ID 479260, 7 pages

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

## The Explicit Expression of the Drazin Inverse and Its Application

^{1}School of Science, Guangxi University for Nationalities, Nanning 530006, China^{2}School 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

#### References

- N. Castro-González, E. Dopazo, and M. F. Martínez-Serrano, “On the Drazin inverse of the sum of two operators and its application to operator matrices,”
*Journal of Mathematical Analysis and Applications*, vol. 350, no. 1, pp. 207–215, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. S. Cvetković-Ilić, D. S. Djordjević, and Y. Wei, “Additive results for the generalized Drazin inverse in a Banach algebra,”
*Linear Algebra and Its Applications*, vol. 418, no. 1, pp. 53–61, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. E. Hartwig, G. Wang, and Y. Wei, “Some additive results on Drazin inverse,”
*Linear Algebra and Its Applications*, vol. 322, no. 1–3, pp. 207–217, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Liu, L. Xu, and Y. Yu, “The representations of the Drazin inverse of differences of two matrices,”
*Applied Mathematics and Computation*, vol. 216, no. 12, pp. 3652–3661, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. S. Cvetković and G. V. Milovanović, “On Drazin inverse of operator matrices,”
*Journal of Mathematical Analysis and Applications*, vol. 375, no. 1, pp. 331–335, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Wang, Y. Wei, and S. Qiao,
*Generalized Inverses: Theory and Computations*, Science Press, Beijing, China, 2004. - C. Y. Deng, “The Drazin inverse of bounded operators with commutativity up to a factor,”
*Applied Mathematics and Computation*, vol. 206, no. 2, pp. 695–703, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Y. Deng, “The Drazin inverses of sum and difference of idempotents,”
*Linear Algebra and Its Applications*, vol. 430, no. 4, pp. 1282–1291, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. L. Campbell and C. D. Meyer Jr.,
*Generalized Inverse of Linear Transformations*, Pitman, Boston, Mass, USA, 1991. View at Zentralblatt MATH · View at MathSciNet - M. P. Drazin, “Pseudo-inverses in associative rings and semigroups,”
*The American Mathematical Monthly*, vol. 65, pp. 506–514, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. D. Meyer Jr. and N. J. Rose, “The index and the Drazin inverse of block triangular matrices,”
*SIAM Journal on Applied Mathematics*, vol. 33, no. 1, pp. 1–7, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet