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Advances in Mathematical Physics
Volume 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

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; moc.361@6761iuhgnoy

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 [710] 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 haveFrom (3), we haveThe 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 obtainFrom (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 obtainBy (14) and (15), we getFrom (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 andSimilarly, is invertible and Since , are nilpotent and , we can conclude that is nilpotent. Thus . Since is Drazin invertible and by (16)-(17), we obtainFrom (18), we getwhereNext, 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 haveSince , we have , where is Binomial coefficients.
By (19) and (21), we obtainwhere and

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

Proof. Firstly, as the proof of Theorem 3, we derive (13). From , we havewhere 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 aswhere is invertible and is nilpotent. By , we obtainTherefore, we getSince , . By the Lemma 5.1.1 of [[15], Chapter V], is nilpotent. Thus, . By Lemma 2, we getwhere Note thatwhere . Since , it implies that . According to (35), we haveSince is invertible, by (8) of Lemma 2 and (35),where , , andBy (37), exists and its expression is given as in (25).
According to Lemma 1 and (ii), we obtainSince , we derivewhere If , by (10) and (40), we havewhere Similarly, we obtainwhere
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 , thenThus, we have andAs (51), we also have Similarly, we deriveFrom (10), (51), and (53), we have Therefore, we getwhere , , and denote the minimum absolute value and nonzero eigenvalues for the matrix-value function and the constant matrix , respectively.
If , thenSimilarly, we haveBy (57) and (58), we have where From (56)-(58), we havewhere . From (19), we deriveBy (62), we getAs (51), we obtain