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Sheng, Xingping

doi:https://doi.org/10.1155/2023/4110507

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4110507 | https://doi.org/10.1155/2023/4110507

Algebraic Perturbation Theorems of Core Inverse and Core-EP Inverse

Xingping Sheng¹

Academic Editor: Predrag S. Stanimirović

Received15 Oct 2022

Revised26 Jan 2023

Accepted02 Feb 2023

Published24 Feb 2023

Abstract

In this paper, the algebraic perturbation theorems of the core inverse and core-EP inverse are discussed for a square singular matrix with different indices, and the expressions of the algebraic perturbation for these two new generalized inverses are presented. As their applications, some properties of the core inverse and core-EP inverse are proven again by using the expressions of their algebraic perturbation. In the last section, two numerical examples are considered to demonstrate the main results.

1. Introduction

In 1974, Bunch and Rose in [1] first used algebraic perturbation methods to solve the nonsingular linear equations. Shortly afterwards, algebraic perturbations of linear operators in Banach spaces were investigated by Rall in [2]. Soon after, singular square matrix was perturbed algebraically to obtain a nonsingular matrix, and the algebraic perturbation theory of Moore–Penrose inverse was also given in [3]. Since then, many scholars have studied algebraic perturbation theories for some other generalized inverses, which can be seen in [4–7].

Throughout the paper, the same notations and terms as in [8, 9] are used. and denote the dimensional complex space and the set of all complex matrices with rank over complex field , respectively, then stands the identity matrix of order . For any , we write the symbols , and for its range, null space, rank, and conjugate transpose, respectively, when , is the inverse of it. For the closed subspaces and , if , denotes a projector (idempotent) on along , and is the orthogonal projector onto . The index of , denoted by , is defined as the minimal nonnegative integer satisfying . The set of square matrices with index one, which is denoted by , is

Definition 1. (see [8]). Let , if satisfies the following four equations:Then, is called Moore–Penrose inverse of , which is denoted by .

Definition 2. (see [8]). For any , if , then the Drazin inverse exists, which is the unique solution of the following three equations:When , the Drazin inverse is called the group inverse of , which is denoted by .
The group inverse is a useful tool in the study of Markov chains and Drazin inverse is used to study the singular differential and difference equations (8).
Let , , and with and . The solution of (4), denoted by , is called the generalized inverse with range and null space of .These concepts for different generalized inverses all can be seen in [8].
In 2010, the core inverse of any was first introduced by Baksalary and Trenkler in [10].

Definition 3. (see [10]). A matrix is called the core inverse of , if it satisfieswhere denotes an orthogonal projection on .
If , its core inverse always exists and unique. The paper [10] attracted a lot of scholars’ attention. Many properties, characterizations, representations, perturbation, and partial ordering of core inverse were presented in [11–16]. Many scholars studied the theories of core inverse in rings, -algebra and Banach space operator, which can be found in [17–21].
The core-EP inverse was originated by Manjunatha Prasad and Mohana in [22] as follows:

Definition 4. (see [22]). For any , if then the core-EP inverse of exists, which is the solution of the following equation:The core inverse and core-EP inverse can be used to study the constrained system of linear equations:for and . If the system (7) is inconsistent, the least squares solution of (7) should be consider, which iswhere is the 2-norm. The unique least squares solution of (7) can be expressed as , when and the solution is [13].
Recently, a number of numerical methods, iterative methods, and theoretical investigations devoted to computation and representation have been presented in [23–29]. The notion of core-EP inverse have been extended to -rings and Hilbert space operators in [30–33]. Until now, the algebraic perturbation theorems of the core inverse and core-EP inverse for a square singular matrix cannot be seen.
Inspired by [5], the purpose of this paper is to establish the algebraic perturbation theorems of core inverse and core-EP inverse. The main algebraic perturbation expressions for core inverse and core-EP inverse are also given. Then, some properties of the core inverse and core-EP inverse are proven again by using their algebraic perturbation results.
The paper is organized as follows: The necessary preliminary results are surveyed in the next section. In Section 3, the algebraic perturbation theories of core inverse and core-EP inverse are proposed. In the last section, two numerical examples are presented to demonstrate the algebraic perturbation theorems.

2. Preliminaries

In this section, some auxiliary definitions and lemmas will be given.

Lemma 1 (see [8]). Let , , and , then the projector satisfies the following:(1), if and only if ,(2), if and only if .

Lemma 2 (see [8]). Let , then .

Lemma 3 (see [8]). Let and , then .

Lemma 4 (see [8]). Let with . If is the full rank decomposition of , then

In [10], Baksalary and Trenkler also gave a useful result of the core inverse of .

Lemma 5 (see [10]). Let , then is core invertible and

In [25], the author introduced the notion of the core-EP decomposition for a matrix with .

Definition 5. (see [25]). If satisfies , then, is the sum of matrices, and , i.e.,wherefurthermore,

3. Main Results

In this section, the algebraic perturbation theorems for the core inverse and core-EP inverse of a square singular matrix with different index are established and the corresponding algebraic perturbation expressions are also given.

First, the algebraic perturbation theory for the core inverse of a matrix is considered.

Theorem 1. Let satisfy . The is a matrix whose columns form a base for , i.e., . If we denote , then(1) is nonsingular.(2).(3).

Proof. (1)For any , if , then Therefore, This means By using Lemma 3, we have , which implies that matrix is nonsingular.(2)For any , denote , then . Furthermore, This implies that , which means If equality (18) is written as the following equation: Following (19) and , we have If equation (18) is rewritten as the following equality: Following the same line, we obtain Thus, from (18), (20), and (22), we have the following equalities: and Hence, from Definition 1.(3)From the result of (2), we haveBy using Lemma 1 and , the following equalities hold:andBy using Lemma 4, we haveIt follows from (26)–(28) thatHence,Applying equation (30) into formula (25), we can getFrom the proof and results of Theorem 1, the following propositions can be obtained, which is consistent with Lemma 5.

Corollary 1. Let with , then the core inverse is a -inverse of with range and nullspace , that is

Furthermore,

Proof. From (18) and the second result of Theorem 1, we haveThis means is a inverse of .
By the (20) and (22), the equation (32) is right.
Applying the properties of and , we haveUsing again, we haveBy direct deduction,Thus,In the next theorem, the algebraic perturbation results of core-EP inverse will be discussed.

Theorem 2. Let satisfy and . If the matrix satisfies , then the following statements for the matrix hold:(1) is nonsingular.(2).(3).

Proof. (1)For any , if , then This means So, From Lemma 3, we know , hence the matrix is nonsingular.(2)For any , if let , then It follows that , we have This implies Hence, further transformations, we have This means Using the same line, we have This yields Following from (45), (47), and (49), and direct verification, Using (47) and (49) again, we have From (51), we get From (50) and (52), the matrix satisfies (6), so .(3)Following the same idea for proving the third result of Theorem 1, we have On the other hand, So, the equality holds from (53) and (54).

Remark 1. When , the algebraic perturbation expressions of the core-EP inverse are just that of the core inverse.
In the next subsection, we will use the algebraic perturbation method of core-EP inverse to prove some properties of core-EP inverse again, which have been proposed in [22, 25].

Corollary 2. Let satisfy the conditions of Definition 5. If is the core-EP decomposition, then(1).(2).

Proof. (1)Let be the core-EP decomposition for the matrix , from the first equality of (12), we know This implies So, From the second result of Theorems 1 and 2, we have According to the condition of and the properties of M-P inverse and Drazin inverse , we have By direct deduction Hence, Similarly, can be proven(2)From (51), we know It follows (50), (52), and (62), and we get From the results of Corollaries 1 and 2, when , the core-EP inverse is exactly the core inverse.

4. Numerical Examples

In this section, two numerical examples of computing the core inverse and core-EP inverse are presented to demonstrate the algebraic perturbation theorems. The first example is taken from [10, 13], while the second example comes from [23, 28].

Example 1. (see [10, 13]). Let us use the same example [9, 12].Solution: take , it is easy to check .
Construct the matrix.then, is nonsingular, and
Last, by directly computing, we have

Example 2. (see [23, 28]). Let us test the following matrix from [23, 28]:Solution:Then,andBy directly computingThen,This yields

5. Conclusion

In this paper, two algebraic perturbation theorems of the core inverse and core-EP inverse for a square singular matrix are proposed. Some properties of the core inverse and core-EP inverse can be proven again by using algebraic perturbation expressions.

Data Availability

The data supporting the findings of this study can be accessed from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the NSF China (grant no. 11901101), the NSF of Anhui Province (grant no. 2008085MA12), and the Projects of Anhui Provincial University Top Talent in Subject and Specialty (grant no. gxbjZD2020072).

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Copyright

Copyright © 2023 Xingping Sheng. 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.

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