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 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 obtainSince (see [[6], Theorem 2] for detail), we haveAccording to (65) (see [6, ,] for detail), thenSimilarly, we also haveFrom (66) and for , or , we obtainSimilarly, we haveBy (64),(68), and (69), we obtain
Theorem 6. Let and be two Drazin invertible matrix-value functions. If , thenFurthermore, we have
Proof. Since and as the proof of Theorem 4, we havewhere . From (i), we haveNote thatAs (66), we have Thus, we obtain Note that and from (76), we haveFor and from Theorem 4 and Lemma 1, we obtainwhere 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 , thenwhere [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 defineFrom 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 easilyFrom (62), we obtainwhere Note that According to , Theorem 3, and Theorem 5, we haveSimilarly, 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 , thenwhere [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 , thenwhere and .
For simplicity, let and be the core-nilpotent decomposition of and in the following result, respectively. We also assume thatThus, 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).