Advances in Mathematical Physics
Volume 2019 (2019), Article ID 8136215, 14 pages
https://doi.org/10.1155/2019/8136215

Research Article

Perturbation Bound for the Drazin Inverse of the Matrix-Value Function

Yonghui Qin,¹ Zhenshu Xie,¹ and Xiaoji Liu²

¹College of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China
²College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China

Correspondence should be addressed to Yonghui Qin; yonghui1676@163.com

Received 8 August 2018; Accepted 1 November 2018; Published 2 January 2019

Academic Editor: Pavel Kurasov

Copyright © 2019 Yonghui Qin 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.

The perturbation analysis of the differential for the Drazin inverse of the matrix-value function is investigated. An upper bound of the Drazin inverse and its differential is also considered. Applications to the perturbation bound for the solution of the matrix-value function coefficients some matrix equations are given.

1. Introduction

The Drazin inverse has been widely applied in the fields, like singular differential equations [1], linear operators theory analysis [2, 3], Markov chains [4], and so on. For the systems with a nonconstant coefficient, one required knowledge of how to differentiate the Drazin inverse as given in [5]. The necessary information for the derivative of the Drazin inverse is developed in [6]. Many results on perturbation analysis for the Drazin inverse of a matrix have been investigated in [7–10] and reference therein.

Let be the set of all complex matrix. For any , there exists a unique matrix such that [1]

is called the Drazin inverse of , denoted by , where is the index of , and denoted by . The core-nilpotent decomposition of is , where is invertible and is a nilpotent matrix and as in [1]. Let be a set of the matrix-value function. For , the definition of the Moore-Penrose inverse for a matrix-valued function is given in [11]. For , the equations in (1) are extended to the Drazin inverse of a matrix-value function in [6]. For , then for all . The Drazin inverse of is defined by with as in [6] and the core-nilpotent decomposition of with is given as follows:

where is a nonsingular matrix-value function, is nonsingular, and is nilpotent for all in (2), respectively. From (2), we easy have the Drazin inverse of the matrix-value function as follows:

As [6], we have

From (3), we have

The basic continuity properties of were developed in [12]. For a differentiable matrix-valued function , we denote by and by , respectively.

The perturbation bound theory of the Drazin inverse on Banach algebra is considered in [13]. For a matrix-value differentiable function, the derivative of the Drazin inverse is defined in [6]. In [12], Campbell et al. investigate the continuity properties of the Drazln inverse. They proved that a sequence of the Drazin inverse of matrix converges to if and only if there exists a such that core-rank = core-rank for . In [5], Campbell et al. considered the applications of the Drazin inverse to the differential equation . In [14], Kelly gave the results on the multivariate, real-valued functions on induced matrix-valued functions on the space of -tuples of pairwise-commuting self-adjoint matrices. Kelly also examined the geometry of this space of matrices and also concluded that the best notion of differentiation of these matrix functions is differentiation along curves in [14]. There are not some results on the perturbation analysis for the differential of the Drazin inverse of the matrix-value function . In this paper, we present some results of perturbation analysis for the differentiable of the Drazin inverse of the matrix-value function with respect to an interval for any complex numbers . We will also consider the perturbation bound theory for the Drazin inverse of a matrix-value function .

2. Preliminaries

Some notations and lemma will be presented in the section.

Lemma 1 (see [6]). Suppose that and are differentiable matrix-valued functions. Let be any integer such that for all . In particular, can be taken to be . Then

From Lemma 1, we derive the following lemma from the results in [1, 15].

Lemma 2. Let

where , , and . If and are Drazin invertible, then and are Drazin invertible,

where

3. Perturbation Analysis for the Drazin Inverse of Matrix-Valued Function

Let be constants and a matrix-value function with all , the perturbation matrix is given as , and the index of is denoted by . For simplicity, let in this paper. We will analyze the perturbation bound for the Drazin inverse of matrix-value function with the condition and , respectively.

For simplicity, we first consider that elements of are linear function, which is given as . Let and be Binomial coefficients.

Theorem 3. Let be matrix-value function with a small perturbation and . Assume that exits and it is differentiable for all . If , then

where , are both core-nilpotent decomposition (see [12]), and

Proof. (i) Since is Drazin invertible, and are presented by (2) and (3), respectively. By and from (2), we obtain and . By (2), we obtain

From (13), we prove that . Now, we will consider the Drazin inverse properties of the matrix-value function . Similar to (2), then and have the block matrix form as follows:

where , are invertible and , are nilpotent. From , , we obtain

By (14) and (15), we get

From (16), we obtain exists exist.
Next, we present the perturbation analysis for the Drazin inverse of , respectively. If is invertible, from , it implies that is invertible. Similarly, is invertible. Since is invertible, is nilpotent, and , which imply that is a nilpotent matrix and

Similarly, is invertible and Since , are nilpotent and , we can conclude that is nilpotent. Thus . Since is Drazin invertible and by (16)-(17), we obtain

From (18), we get

where

Next, we consider the derivative of the polynomial matrix with respect to . As [12], let be the core-nilpotent decomposition of the matrix-valued function , where is invertible and is nilpotent, respectively. Since the elements of the matrix are polynomials, we have is differentiable and and are also. To this end, firstly we consider the expression of the derivative . Note that . By Lemma 1, we have

Since , we have , where is Binomial coefficients.
By (19) and (21), we obtain

where and

Theorem 4. Let be an matrix-valued function with a small scalar and . If , then is Drazin invertible and

Furthermore, if differentiable, then

where

Proof. Firstly, as the proof of Theorem 3, we derive (13). From , we have

where is invertible and is nilpotent. Thus, we prove that

Therefore, we get that

From Lemma 2, it implies that and are both Drazin invertible. Note that is invertible. Here, we need to consider the Drazin invertible properties of and its expression. Since and by the Theorem 5.1.3 of [[15], Chapter V], we prove that is Drazin invertible. As (2), and can be rewritten as

where is invertible and is nilpotent. By , we obtain

Therefore, we get

Since , . By the Lemma 5.1.1 of [[15], Chapter V], is nilpotent. Thus, . By Lemma 2, we get

where Note that

where . Since , it implies that . According to (35), we have

Since is invertible, by (8) of Lemma 2 and (35),

where , , and

By (37), exists and its expression is given as in (25).
According to Lemma 1 and (ii), we obtain

Since , we derive

where

If , by (10) and (40), we have

where

Similarly, we obtain

where
According to the result (10) of (i) and (39), (40), (42), and (44), we have the following simplification:

4. Perturbation Bound for the Drazin Inverse of the Matrix-Value Function

In this subsection, we consider the perturbation bound for the Drazin inverse and its differential of the matrix-value function with any in .

Theorem 5. Let , be both Drazin invertible matrix-value function and . Then

Furthermore, we obtain

where , , and

Proof. (i) As Theorem 3, let . If , then

Thus, we have and

As (51), we also have

Similarly, we derive

From (10), (51), and (53), we have

Therefore, we get

where , , and denote the minimum absolute value and nonzero eigenvalues for the matrix-value function and the constant matrix , respectively.
If , then

Similarly, we have

By (57) and (58), we have

where

From (56)-(58), we have

where . From (19), we derive

By (62), we get

As (51), we obtain

Since (see [[6], Theorem 2] for detail), we have

According to (65) (see [6, ,] for detail), then

Similarly, we also have

From (66) and for , or , we obtain

Similarly, we have

By (64),(68), and (69), we obtain

Theorem 6. Let and be two Drazin invertible matrix-value functions. If , then

Furthermore, we have

Proof. Since and as the proof of Theorem 4, we have

where . From (i), we have

Note that

As (66), we have

Thus, we obtain

Note that

and from (76), we have

For and from Theorem 4 and Lemma 1, we obtain

where and It is proved by (iii) of Theorem 5.

5. Applications

In the section, we state applications of our results given in Section 4 to the perturbation bound for three matrix equations.

5.1. For Case

We first consider the following equation [16]:

where is a matrix-value function with for all and is given and is unknown. If the range space , then

where [16]. Assume that there are perturbation constant matrices and such that and are both Drazin invertible. Thus, there exists a matrix-value function with , which satisfies

For simplicity, in the following results, we define and . We also assume that are given in Theorem 5 and , and define

From Theorem 3 and Theorem 5, we have the following result.

Theorem 7. Let and satisfy (82). Assume that . If and , then

Proof. If , then satisfies (84). The following proof can be derived from Theorem 3 and Theorem 5. Note that easily

From (62), we obtain

where Note that According to , Theorem 3, and Theorem 5, we have

Similarly, by Theorem 4 and Theorem 6, we obtain the following results.

Theorem 8. Let and satisfy (82) with and . Assume that . If and , then

5.2. For Case

In the following, we consider the solutions for the equation as follows:

where and are given in (90). If the null space , then

where [16]. As (84), we have

The following result can derived from Theorem 3 and Theorem 5.

Theorem 9. Let and satisfy (90). Assume that . If and satisfy (92), then

The following result can derived from Theorem 4 and Theorem 6.

Theorem 10. Let and satisfy (90). Assume that . If and , then

5.3. For Case

Similarly, we consider the equation as follows:

where with , with and are all given, , and is unknown. If the range space and the null space , then

where and .

For simplicity, let and be the core-nilpotent decomposition of and in the following result, respectively. We also assume that

Thus, we have the following result as Theorems 9 and 10.

Theorem 11. Let and satisfy the equation (94). Assume that

If and , then

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11361009, 61772006, 11561015, and 11701119), the Special Fund for Science and Technological Bases and Talents of Guangxi (Grant no. 2016AD05050), the Special Fund for Bagui Scholars of Guangxi (Grant no. 2016A17) and High Level Innovation Teams and Distinguished Scholars in Guangxi Universities, and the open fund of Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis (no. HCIC201607).

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