# Algebraic Perturbation Theorems of Core InverseA㊮ and Core-EP Inverse A㊯

Authors: Xingping Sheng
Journal: Journal of Mathematics
 (2023)

---

## Abstract

In this paper, the algebraic perturbation theorems of the core inverseA㊮ and core-EP inverse A㊯ are discussed for a square singular matrix A 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 A㊮ and core-EP inverse A㊯ 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.


---

## Body

## 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. ℂn and ℂrm×n denote the n dimensional complex space and the set of all m×n complex matrices with rank r over complex field ℂ, respectively, then In stands the identity matrix of order n. For any A∈ℂrm×n, we write the symbols ℛA,NA,rA, and A∗ for its range, null space, rank, and conjugate transpose, respectively, when r=m=n, A−1 is the inverse of it. For the closed subspaces L and M, if L⊕M=ℂn, PL,M denotes a projector (idempotent) on L along M, and PL is the orthogonal projector onto L. The index of A∈ℂn×n, denoted by indA=k, is defined as the minimal nonnegative integer k satisfying rAk+1=rAk. The set of square matrices with index one, which is denoted by ℂnCM, is(1)ℂnCM=A∈ℂn×n|rA2=rA=A∈ℂn×n|indA=1.Definition 1.
(see [8]). Let A∈ℂm×n, if X∈ℂn×m satisfies the following four equations:(2)AXA=A,XAX=X,AX∗=AX,XA∗=XA.
Then,X is called Moore–Penrose inverse of A, which is denoted by A†.Definition 2.
(see [8]). For any A∈ℂn×n, if indA=k, then the Drazin inverse AD exists, which is the unique solution of the following three equations:(3)AkXA=Ak,XAX=X,AX=XA.
WhenA∈ℂnCM, the Drazin inverse is called the group inverse of A, which is denoted by A#.
The group inverseA# is a useful tool in the study of Markov chains and Drazin inverse AD is used to study the singular differential and difference equations (8).
LetA∈ℂrm×n, T⊆ℂn, and S⊆ℂm with dimT=s≤r and dimS=m−s. The solution of (4), denoted by AT,S2, is called the generalized inverse with range T and null space S of A.(4)XAX=X,RX=T,NX=S.
These concepts for different generalized inverses all can be seen in [8].
In 2010, the core inverse of anyA∈ℂnCM was first introduced by Baksalary and Trenkler in [10].Definition 3.
(see [10]). A matrix A㊮∈ℂn×n is called the core inverse of A∈ℂnCM, if it satisfies(5)AA㊮=PℛA,ℛA㊮⊆ℛA,where PℛA=AA† denotes an orthogonal projection on ℛA.
IfA∈ℂnCM, 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, C∗-algebra and Banach space operator, which can be found in [17–21].
The core-EP inverseA㊯ was originated by Manjunatha Prasad and Mohana in [22] as follows:Definition 4.
(see [22]). For any A∈ℂn×n, if indA=k then the core-EP inverse X=A㊯ of A exists, which is the solution of the following equation:(6)XAX=X,ℛX=ℛX∗=ℛAk.
The core inverseA㊮ and core-EP inverse A㊯ can be used to study the constrained system of linear equations:(7)Ax=b,x∈ℛAk,for A∈ℂn×n and b∈ℂn,k=indA. If the system (7) is inconsistent, the least squares solution of (7) should be consider, which is(8)minx∈ℛAkb−Ax2,where ⋅2 is the 2-norm. The unique least squares solution of (7) can be expressed as A㊯b, when indA=1 and the solution is A㊮b [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 A㊮ and core-EP inverse A㊯ for a square singular matrix A 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 Section3, 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]).
LetL⊆ℂn, M⊆ℂn, and L⊕M=ℂn, then the projector PL,M satisfies the following:(1)
PL,MA=A, if and only if ℛA⊆L,(2)
APL,M=A, if and only if NA⊇M.Lemma 2 (see [8]).
LetA∈ℂrm×n, then ℛA=NA∗⊥.Lemma 3 (see [8]).
LetA∈ℂrn×n and indA=k, then ℛAk⊕NAk=ℂn.Lemma 4 (see [8]).
LetA∈ℂrn×n with indA=1. If A=FG is the full rank decomposition of A, then(9)A#=FGF−2G.In [10], Baksalary and Trenkler also gave a useful result of the core inverse of A∈ℂnCM.Lemma 5 (see [10]).
LetA∈ℂnCM, then A is core invertible and(10)A㊮=A#AA†.In [25], the author introduced the notion of the core-EP decomposition for a matrix A∈ℂrn×n with indA=k.Definition 5.
(see [25]). If A∈ℂrn×n satisfies indA=k, then, A is the sum of matrices, A1 and A2, i.e.,(11)A=A1+A2,where(12)A1=AkAk†A,A2=In−AkAk†A,furthermore,(13)A1∈ℂnCM,A2k=0,A1∗A2=A2A1=0.


## 3. Main Results

In this section, the algebraic perturbation theorems for the core inverseA㊮ and core-EP inverse A㊯ of a square singular matrix A with different index are established and the corresponding algebraic perturbation expressions are also given.First, the algebraic perturbation theory for the core inverseA㊮ of a matrix A∈ℂnCM is considered.Theorem 1.
LetA∈ℂrn×n satisfy indA=1. The B∈ℂn−rn×n−r is a matrix whose columns form a base for NA∗, i.e., ℛB=NA∗. If we denote H=A+BB∗, then(1)
H is nonsingular.(2)
A㊮=H−1AH−1.(3)
A㊮=H−1−PNA,ℛABB∗#.Proof.
(1)
For anyx∈ℂn, if Hx=A+BB∗x=0, then(14)Ax=−BB∗x∈ℛA∩ℛBB∗=ℛA∩ℛB=ℛA∩NA∗=0.Therefore,(15)Ax=0,BB∗x=0.This means(16)x∈NA∩NBB∗=NA∩NB∗=NA∩ℛB⊥=NA∩ℛA.By using Lemma 3, we havex=0, which implies that matrix H is nonsingular.(2)
For anyx∈ℂn, denote z=H−1Ax, then Hz=Ax.Furthermore,(17)Ax−z=BB∗z∈ℛA∩ℛBB∗=ℛA∩ℛB=ℛA∩NA∗=0.This implies thatAx=Az=AH−1Ax, which means(18)A=AH−1A.If equality (18) is written as the following equation:(19)0=A−AH−1A=AH−1H−A=AH−1BB∗.Following (19) and rBB∗=rB=n−r=n−rA=n−rAH−1, we have(20)NAH−1=ℛBB∗=ℛB=NA∗.If equation (18) is rewritten as the following equality:(21)0=A−AH−1A=H−AH−1A=BB∗H−1A.Following the same line, we obtain(22)ℛH−1A=NBB∗=NB∗=ℛA.Thus, from (18), (20), and (22), we have the following equalities:(23)AH−1AH−1=AH−1=PℛAH−1,NAH−1=PℛA,NA∗=PℛA,and(24)ℛH−1AH−1=ℛH−1A=ℛA.Hence,A㊮=H−1AH−1 from Definition 1.(3)
From the result of (2), we have(25)A㊮=H−1AH−1=H−1H−BB∗H−1=H−1−H−1BB∗H−1.
By using Lemma 1 andℛB=NA∗, the following equalities hold:(26)APNA,ℛA=0,and(27)PNA,ℛAB=0.
By using Lemma 4, we have(28)BB∗#=BB∗B−2B∗.
It follows from (26)–(28) that(29)HPNA,ℛABB∗#H=A+BB∗PNA,ℛABB∗B−2B∗A+BB∗=BB∗BB∗B−2B∗BB∗=B∗B.
Hence,(30)H−1BB∗H−1=PNA,ℛABB∗#.
Applying equation (30) into formula (25), we can get(31)A㊮=H−1−PNA,ℛABB∗#.
From the proof and results of Theorem 1, the following propositions can be obtained, which is consistent with Lemma 5.Corollary 1.
LetA∈ℂnCM with rA=r, then the core inverse A㊮ is a 2-inverse of A with range ℛA and nullspace NA∗, that is(32)A㊮=AℛA,NA∗2.Furthermore,(33)A㊮=H−1AH−1=A#AA†.Proof.
From (18) and the second result of Theorem 1, we have(34)A㊮AA㊮=H−1AH−1AH−1AH−1=H−1AH−1=A㊮.
This meansA㊮ is a 2 inverse of A.
By the (20) and (22), the equation (32) is right.
Applying the properties ofA# and A†, we have(35)ℛA#=ℛA,NA†=NA∗.
UsingℛB=NA∗ again, we have(36)BB∗A#=0,A†BB∗=0.
By direct deduction,(37)HA#AA†H=A+BB∗A#AA†A+BB∗=AA#AA†A=A.
Thus,(38)A㊮=H−1AH−1=A#AA†.
In the next theorem, the algebraic perturbation results of core-EP inverse will be discussed.Theorem 2.
LetA∈ℂrn×n satisfy indA=k and rAk=s≤r. If the matrix B∈ℂn−sn×n−s satisfies ℛB=NAk∗, then the following statements for the matrix T=PℛAkA+BB∗ hold:(1)
T is nonsingular.(2)
A㊯=T−1PℛAkAT−1.(3)
A㊯=T−1−PNAk,ℛAkBB∗#.Proof.
(1)
For anyx∈ℂn, if Tx=0, then(39)PℛAkAx=−BB∗x∈ℛAk∩ℛBB∗=ℛAk∩ℛB=ℛAk∩ℛAk∗=0.This means(40)PℛAkAx=0,BB∗x=0.So,(41)x∈NA∩NBB∗⊂NAk∩NB∗=NAk∩ℛAk.From Lemma 3, we knowx=0, hence the matrix T is nonsingular.(2)
For anyx∈ℂn, if let z=T−1PℛAkAx, then(42)Tz=PℛAkAx.It follows thatT=PℛAkA+BB∗, we have(43)PℛAkAx−z=BB∗z∈ℛAk∩ℛBB∗=ℛAk∩ℛAk⊥=0.This implies(44)PℛAkAx=PℛAkAz=PℛAkAT−1PℛAkAx.Hence,(45)PℛAkA=PℛAkAT−1PℛAkA.further transformations, we have(46)PℛAkA=PℛAkPℛAkA+BB∗=PℛAkT.This means(47)PℛAk=PℛAkAT−1.Using the same line, we have(48)PℛAkA=PℛAkA+BB∗PℛAk=TPℛAk.This yields(49)PℛAk=T−1PℛAkA.Following from (45), (47), and (49), and direct verification,(50)XAX=T−1PℛAkAT−1AT−1PℛAkAT−1=T−1PℛAkAT−1PℛAkAT−1=T−1PℛAkAT−1=X.Using (47) and (49) again, we have(51)X=T−1PℛAkAT−1=PℛAkT−1=T−1PℛAk.From (51), we get(52)ℛX=ℛPℛAkT−1=ℛAk=ℛPℛAkT−1∗=ℛT−1PℛAk∗=ℛX∗.From (50) and (52), the matrix X=T−1PℛAkAT−1 satisfies (6), so X=A㊯.(3)
Following the same idea for proving the third result of Theorem 1, we have(53)A㊯=T−1PℛAkAT−1=T−1T−BB∗T−1=T−1−T−1BB∗T−1.On the other hand,(54)TPNAk,ℛAkBB∗#T=PℛAkA+BB∗PNAk,ℛAkBB∗B−2B∗PℛAkA+BB∗=BB∗BB∗B−2B∗BB∗=BB∗.So, the equalityA㊯=T−1−PNAk,ℛAkBB∗# holds from (53) and (54).Remark 1.
Whenk=1, 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.
LetA satisfy the conditions of Definition 5. If A=A1+A2 is the core-EP decomposition, then(1)
A㊯=A1㊮=ADAkAk†=AkAk+1†.(2)
A㊯=AℛAk,NAk∗2.Proof.
(1)
LetA=A1+A2 be the core-EP decomposition for the matrix A, from the first equality of (12), we know(55)A1=AkAk†A=PℛAkA.This implies(56)ℛA1=ℛPℛAkA=ℛAk.So,(57)ℛB=ℛAk⊥=ℛA1⊥=NA1∗.From the second result of Theorems 1 and 2, we have(58)A㊯=T−1PℛAkAT−1=T−1A1T−1=A1㊮.According to the condition ofB and the properties of M-P inverse Ak㊯ and Drazin inverse AD, we have(59)B∗AD=0,Ak†B=0.By direct deduction(60)TADAkAk†T=PℛAkA+BB∗ADAkAk†PℛAkA+BB∗=PℛAkAADAkAk†PℛAkA=PℛAkAkAk†PℛAkA=PℛAkA.Hence,(61)A㊯=T−1PℛAkAT−1=ADAkAk†.Similarly,A㊯=T−1PℛAkAT−1=AkAk+1† can be proven(2)
From (51), we know(62)NX=NT−1PℛAk=NPℛAk=NAk∗.It follows (50), (52), and (62), and we get(63)A㊯=AℛAk,NAk∗2.From the results of Corollaries 1 and 2, whenindA=1, the core-EP inverse is exactly the core inverse.


## 4. Numerical Examples

In this section, two numerical examples of computing the core inverseA㊮ and core-EP inverse A㊯ 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].(64)A=0100100−12,indA=1,rA=2.
Solution: takeB=−110, it is easy to check ℛB=NA∗.
Construct the matrix.(65)H=A+BB∗=100−1200−12,then, H is nonsingular, and H−1=1001/21/201/21/21
Last, by directly computing, we have(66)A㊮=H−1AH−1=10012120121210100100−121001212012121=1212012120141412.Example 2.
(see [23, 28]). Let us test the following matrix from [23, 28]:(67)A=3−1−1−111−31111−3−13−1−1.
Solution:(68)A3=16−1616−16−1616−161616−1616−16−1616−1616,rA3=1,indA=3.
Then,(69)PℛA3=14−1414−14−1414−141414−1414−14−1414−1414,and(70)B=1−11100010001.
By directly computing(71)T=PℛAkA+BB∗=14−1414−14−1414−141414−1414−14−1414−14143−1−1−111−31111−3−13−1−1+1−111000100011100−10101001=400002−110−12−101−12.
Then,(72)T−1=1400003414−1401434140−141434.
This yields(73)A㊯=T−1PℛAkAT−1=1400003414−1401434140−14143414−1414−14−1414−141414−1414−14−1414−14143−1−1−111−31111−3−13−1−11400003414−1401434140−141434=116−116116−116−116116−116116116−116116−116−116116−116116.


## 5. Conclusion

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



---

*Source: 4110507-2023-02-24.xml*