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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Weak group inverse

Hongxing Wang
  • Corresponding author
  • School of Mathematics, Southeast University, Nanjing, 210096, China and School of Science, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning, 530006, China
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/ Jianlong Chen
Published Online: 2018-10-31 | DOI: https://doi.org/10.1515/math-2018-0100

Abstract

In this paper, we introduce the weak group inverse (called as the WG inverse in the present paper) for square complex matrices of an arbitrary index, and give some of its characterizations and properties. Furthermore, we introduce two orders: one is a pre-order and the other is a partial order, and derive several characterizations of the two orders. The paper ends with a characterization of the core EP order using WG inverses.

Keywords: Group inverse; Weak group inverse; WG order; CE partial order; Core-EP decomposition

MSC 2010: 15A09; 15A57; 15A24

1 Introduction

In this paper, we use the following notations. The symbol ℂm,n is the set of m × n matrices with complex entries; A*, (A) and rk(A) represent the conjugate transpose, range space (or column space) and rank of A ∈ ℂm,n, respectively. Let A ∈ ℂn,n be singular, the smallest positive integer k satisfying rk (Ak+1) = rk (Ak) called the index of A and is denoted by Ind(A). The index of a non-singular matrix A is 0 and the index of a null matrix is 1. The symbol nCM stands for a set of n × n matrices of index less than or equal to 1. The Moore-Penrose inverse of A ∈ ℂm,n is defined as the unique matrix X ∈ ℂn,m satisfying the equations:

(1)AXA=A,(2)XAX=X,(3)(AX)=AX,(4)(XA)=XA,

and is denoted as X = A; PA stands for the orthogonal projection PA = AA. A matrix X such that AXA = A is called a generalized inverse of A. The Drazin inverse of A ∈ ℂn,n is defined as the unique matrix X ∈ ℂn,n satisfying the equations

(6k)XAk+1=Ak,(2)XAX=X,(5)AX=XA,

and is usually denoted as X = AD, where k = Ind(A). In particular, when AnCM, the matrix X is called the group inverse of A, and is denoted as X = A# (see [1]). The core inverse of AnCM is defined as the unique matrix X ∈ ℂn,n satisfying

AX=AA,(X)(A)

and is denoted as X=A# [2]. When AnCM, we call it a core invertible (or group invertible) matrix.

Several generalizations of the core inverse have been introduced, for example, the DMP inverse[3] the BT inverse[4] and the core-EP inverse[5], etc. Let A ∈ ℂn,n with Ind (A) = k. The DMP inverse of A is Ad,† = ADAA [3]. The BT inverse of A is A = (A2A) [4, Definition 1]. The core-EP inverse of A is A=AkAkAk+1Ak [5, Theorem 3.5 and Remark 2]. It is evident that A#=A=Ad,=A in case of AnCM. More results on the core inverse and related problems can be seen in [610].

Furthermore, it is known that the index of a group invertible matrix is less than or equal to 1, that is, a matrix is core invertible if and only if it is group invertible. Although the generalizations of the core inverse have attracted huge attention, the generalizations of group inverse have not received the same kind of attention. Therefore, it is of interest to inquire whether one can do something similar to the group inverse and that too by using some matrix decompositions as a tool as it has been used to study generalizations of core inverse.

In this paper, our main tool is the core-EP decomposition. By using this decomposition, we introduce a generalization of the group inverse for square matrices of an arbitrary index. We also give some of its characterizations, properties and applications.

2 Preliminaries

In this section, we present some preliminary results.

Lemma 2.1 ([1])

Let A ∈ ℂn,n with Ind(A) = k. Then

AD=Ak(Ak+1)#.(1)

The following decomposition is attributed to Hartwig and Spindelböck[11] and is called Hartwig-Spindelböck decomposition

Lemma 2.2

([11, Hartwig-Spindelböck Decomposition]). Let A ∈ ℂn,n with rk(A) = r. Then there exists a unitary matrix U such that

A=U[ΣKΣL00]U,(2)

where Σ = diag(σ1Ir1,σ2Ir2,. . ., σtIr1) is the diagonal matrix of singular values of A, σ1 > σ2 > … ≥ σt > 0, r1 + r2 + … + rt = r, and K ∈ ℂr,r, L ∈ ℂr,n−r satisfy KK* + LL* = Ir.

Furthermore, A is core invertible if and only if K is non-singular, [2]. When AnCM, it is easy to check that

A#=UT1000U,(3)

A#=U[T1T2S00]U,(4)

where T = ΣK and S = ΣL.

The core-nilpotent decomposition of a square matrix is widely used in matrix theory [1, 12] and just to remind ourselves it is given as:

Lemma 2.3

([12, Core-nilpotent Decomposition]). Let A ∈ ℂn,n with Ind(A) = k, then A can be written as the sum of matrices Â1 and Â2, i.e. A = Â1 + Â2, where

A^1nCM,A^2k=0andA^1A^2=A^2A^1=0.

Very similar to core-nilpotent decomposition is the core-EP decomposition of a square matrix of arbitrary index and was introduced by Wang [13]. We record it as:

Lemma 2.4

([13, Core-EP Decomposition]). Let A ∈ ℂn,n with Ind(A) = k, then A can be written as the sum of matrices A1 and A2, i.e. A = A1 + A2, where

  • (i) A1nCM;

  • (ii) A2k=0;

  • (iii) A1A2=A2A1=0.

Here one or both of A1 and A2 can be null.

Lemma 2.5 ([13])

Let the core-EP decomposition of A ∈ ℂn,n be as in Lemma 2.4. Then there exists a unitary matrix U such that

A1=U[TS00]U,A2=U[000N]U,(5)

where T is non-singular, and N is nilpotent. Furthermore, the core-EP inverse of A is

A=UT1000U.(6)

3 WG inverse

In this section, we apply the core-EP decomposition to introduce a generalized group inverse (i.e. the WG inverse) and consider some characterizations of the generalized inverse.

3.1 Definition and properties of the WG inverse

Let A ∈ ℂn,n with Ind(A) = k, and consider the system of equations1

2 AX2=X,3c AX=AA.(7)

Theorem 3.1

The system of equations (7) is consistent and has a unique solution

X=U[T1T2S00]U.(8)

Proof

Let A ∈ ℂn,n with Ind(A) = k. Since A=AkAkAk+1Ak,RAARA. Therefore, (3c) is consistent. Let A be as in (5). From (6), we obtain

A2A=UT1T2S00U(9)

and

AA2A=AA,(10)

that is, A2A is a solution to (3c).

It is obvious that (2′) is consistent. Applying (9), we have

AA2A2=A2A,(11)

that is, A2A is a solution to (2′).

Therefore, from (9), (10) and (11), we derive that (7) is consistent and (8) is a solution of (7).

Furthermore, suppose that both X and Y satisfy (7), then

X=AX2=AAX=AAA=AAY=AY2=Y,

that is, the solution to the system of equations (7) is unique. □

Definition 3.2

Let A ∈ ℂn,n be a matrix of index k. The WG inverse of A, denoted as AW, is defined to be the solution to the system (7).

Remark 3.3

When AnCM, we have AW=A#.

Remark 3.4

In [14, Definition 1], the notion of weak Drazin inverse was given: let A ∈ ℂn,n and Ind(A) = k, then X is a weak Drazin inverse of A if X satisfies (6k). Applying (8), it is easy to check that the WG inverse AW is a weak Drazin inverse of A.

Remark 3.5

Let A ∈ ℂn,n. Applying Theorem 3.1, it is easy to check AWAAW=AW and RAW=RAk.

More details about the weak Drazin inverse can be seen in [1416].

In the following example, we explain that the WG inverse is diσerent from the Drazin, DMP, core-EP and BT inverses.

Example 3.6

Let A=[1010010100010000]. It is easy to check that Ind(A) = 2, the Moore-Penrose inverse A and the Drazin inverse AD are

A=[0.500001100.50000010]andAD=[1011010100000000],

the DMP inverse Ad,† and the BT inverse A are

Ad,=ADAA=[1010010000000000]andA=(A2A)=[0.500001000.50000000],

and the core-EP inverse A and the WG inverse AW are

A=1000010000000000and AW=1010010100000000.

3.2 Characterizations of the WG inverse

Theorem 3.7

Let A ∈ ℂn,n be as in (5). Then

AW=A1#=UT1T2S00U.(12)

Proof

Let A = Â1 + Â2 be the core-nilpotent decomposition of A ∈ ℂn,n. Then AD=A^1#. Applying Lemma 2.4, (5) and (8), we derive (12).

Theorem 3.8

Let A ∈ ℂn,n with Ind(A) = k. Then

AW=AAA#=A2A=A2A.(13)

Proof

Let A be as in (5). Then

AAA=UTS0NT1000TS0NU=UTS00U,A2=UT1000U2=UT2000U,A2=UT2TS+SN0N2U=UT2000U.

It follows from Theorem 3.7 that

AAA#=UTS00U#=UT1T2S00U=AW,A2A=A2A=UT2000TS0NU=UT1T2S00U=AW.

Therefore, we obtain (13). □

Theorem 3.9

Let A ∈ ℂn,n with Ind(A) = k. Then

AW=AkAk+2#A=A2PAkA.(14)

Proof

Let A be as in (5). Then

Ak=U[TkΦ00]U,(15)

where Φ=i=1kTi1SNki. It follows that

AkAk+2#A=UTkΦ00T(k+2)000TS0NU=UT1T2S00U=AW,(16)

PAk=AkAk=UIrk(Ak)000U,A2PAkA=UT2000TS0NU=AW.(17)

Therefore, we have (14). □

It is known that the Drazin inverse is one generalization of the group inverse. We will see the similarities and diσerences between the Drazin inverse and the WG inverse from the following corollaries.

Corollary 3.10

Let A ∈ ℂn,n with Ind(A) = k. Then

rkAW=rkAD=rkAk.

It is well known that (A2)D = (AD)2, but the same is not true for the WG inverse. Applying the core-EP decomposition (5) of A, we have

A2=U[T2TS+SN0N2]U(18)

and

A2W=UT2T4TS+SN00U,AW2=UT2T3S00U.(19)

Therefore, A2W=AW2 if and only if T−4(TS + SN) = T−3S. Since T is invertible, we derive the following Corollary 3.11.

Corollary 3.11

Let A ∈ ℂn,n be as in (5). Then A2W=AW2 if and only if SN = 0.

The commutativity is one of the main characteristics of the group inverse. The Drazin inverse too has the characteristic. It is of interest to inquire whether the same is true or not for the WG inverse. Applying the core-EP decomposition (5) of A, we have

AAW=UTS0NT1T2S00U=UIT1S00U;(20a)

AWA=UT1T2S00TS0NU=UIT1S+T2SN00U.(20b)

Therefore, we have the following Corollary 3.12.

Corollary 3.12

Let the core-EP decomposition of A ∈ ℂn,nbe as in (5). Then AAW=AWA if and only if SN = 0.

Corollary 3.13

Let A ∈ ℂn,n with Ind(A) = k, the core-EP decomposition of A be as in (5) and SN = 0. Then

AW=AD=Ak+1#Ak=At+1At,

where t is an arbitrary positive integer.

Proof

Let the core-EP decomposition of A ∈ ℂn,n be as in (5).

By applying SN = 0 and Ind(A) = k, we have

Ak1=U[Tk1Tk2S0Nk1]U,Ak=U[TkTk1S00]U,Ak+1=U[Tk+1TkS00]U.

It follows from applying (1), (4) and (6) that

Ak+1#=Ak+1#=UT(k+1)T(k+2)S00U,AD=Ak+1#Ak=UT(k+1)T(k+2)S00TkTk1S00U=UT1T2S00U=AW.

Therefore, AW=AD=Ak+1#Ak.

Let t be an arbitrary positive integer. By applying SN = 0, we have

At=U[TtTt1S0Nt]U,At+1=U[Tt+1TtS0Nt+1]U.

It follows from Lemma 2.5 that

At+1=UT(t+1)000U,At+1At=UT(t+1)000TtTt1S0NtU=AW,(21)

Therefore, we derive AW=At+1At, in which t is an arbitrary positive integer. □

4 Two orders

Recall the definitions of the minus partial order, sharp partial order, Drazin order and core-nilpotent partial order [12] :

AB:A,Bm,n,rk(BA)=rk(B)rk(A),(22)

A#B:A,BnCM,A2=AB=BA,(23)

ADB:A,Bn,n,A^1#B^1,(24)

A#,B:A,Bn,n,A^1#B^1andA^2B^2,(25)

in which A = Â1 + Â2 and B=B^1+B^2 are the core-nilpotent decompositions of A and B, respectively. Similarly, in this section, we apply the core-EP decomposition to introduce two orders: one is the WG order and the other is the CE partial order.

4.1 WG order

Consider the binary relation:

AWGB:A,Bn,n,ifA1#B1,(26)

in which A = A1 + A2 and B = B1 + B2 are the core-EP decompositions of A and B, respectively.

Reflexivity of the relation is obvious. Suppose AWGB and BWGC, in which A = A1 + A2, B = B1 + B2 and C = C1 + C2 are the core-EP decompositions of A, B and C, respectively. Then A1#B1 and B1#C1. Therefore A1#C1. It follows from (26) that AWGC.

Example 4.1

Let

A=[111001000],B=[111002000]

Although AWGB and BWGA, AB. Therefore, the anti-symmetry of the binary operation (26) does not hold in general.

Therefore, we have the following Theorem 4.2.

Theorem 4.2

The binary relation (26) is a pre-order. We call this pre-order the weak-group (WG for short) order.

Remark 4.3

The WG order coincides with the sharp partial order onnCM.

We give below two examples to show that WG order is diσerent from Drazin order and that either of two orders does not imply the other order.

Example 4.4

Let A and B be as in Example 4.1. We have

AD=[112000000].

It is easy to check that AWGB.

Since ADAAD, we derive ADB. Therefore, the WG order does not imply the Drazin order.

Example 4.5

Let

A^=[100000000],B^=[100001000],P=[120010001],A=PA^P1=[120000000],B=PB^P1=[122000000],A1=[120000000],A2=0,B1=[122000000],B2=[000001000],

in which A = A1 + A2 and B = B1 + B2 are the core-EP decompositions of A and B, respectively. Then ADB and A1#B1. Therefore, the Drazin order does not imply the WG order.

It is well known that ADBA2DB2, but the same is not true for the WG order as the following example shows:

Example 4.6

Let A and B be as in Example 4.1, then

A2=[111000000],B2=[113000000].

We derive A2WGB2. Therefore, AWGBA2WGB2.

Theorem 4.7

Let A, B ∈ ℂn,n. Then AWGB if and only if there exists a unitary matrix U^ such that

A=U^[TS^1S^20N11N120N21N22]U^,(27a)

B=U^[TS^1T1S^1T1S^2T1S^1S10T1S100N2]U^,(27b)

where T and T1 are invertible, [N11N12N21N22] and N2 are nilpotent.

Proof

Assume AWGB. Let A = A1 + A2 and B = B1 + B2 be the core-EP decompositions of A and B, A1 and A2 be as given in (5), and partition

UB1U=[B11B12B21B22].(28)

Applying (12) gives

B1A1#=U[B11B12B21B22][T1T2S00]U=U[B11T1B11T2SB21T1B21T2S]U.

Since AWGB,A1#B1. It follows from A1A1#=B1A1# that

B11=TandB21=0.(29)

By applying (12) and (29), we have

A1#A1=U[IT1S00]U,A1#B1=U[IT1B12+T2SB2200]U.

It follows from A1#A1=A1#B1 that

T1(ST1SB22B12)=0.

Therefore,

B12=ST1SB22,(30)

in which B22 is an arbitrary matrix of an appropriate size. From (29) and (30), we obtain

B1=U[TST1SB220B22]U.(31)

Since B1 is core invertible and T is non-singular,B22 is core invertible. Let the core-EP decomposition of B22 be as

B22=U1[T1S100]U1,(32)

where T1 is invertible. Denote

U^=U[I00U1].

It is easy to see that U^ is a unitary matrix. Let SU1 be partitioned as follows:

SU1=[S^1S^2],

where the number of columns of S^1 coincides with the size of the square matrix T1. Then

A1=U^[TS^1S^2000000]U^(33)

and

B1=U[TST1SB220U1[T1S100]U1]U=U[I00U1][TSU1T1SU1U1B22U10[T1S100]][I00U1]U=U^[T[S^1S^2]T1[S^1S^2][T1S100]0[T1S100]]U^=U^[TS^1T1S^1T1S^2T1S^1S10T1S1000]U^.(34)

From (26), (33) and (34), we derive (27a) and (27b). □

4.2 CE partial order

Consider the binary relation:

ACEB:A,Bn,n,A1#B1andA2B2,(35)

in which A = A1 + A2 and B = B1 + B2 are the core-EP decompositions of A and B, respectively.

Definition 4.8

Let A, B ∈ ℂn,n. We say that A is below B under the core-EP-minus (CE for short) order if A and B satisfy the binary relation (35).

When A is below B under the CE order, we write ACEB.

Remark 4.9

According to (26) and (35) we derive that the CE order implies the WG order, that is,

ACEBAWGB.(36)

Furthermore,

ACEBAWGBandA2B2.(37)

Theorem 4.10

The CE order is a partial order.

Proof

Reflexivity is trivial.

Let ACEB, BCEC and A = A1 + A2,B = B1 + B2 and C = C1 + C2 are the core-EP decompositions of A, B and C, respectively. Then A1#B1, B1#C1 and A2B2, B2C2. Therefore A1#C1 and A2C2. It follows from Definition 4.8 that ACEC

If ACEB and BCEA, Then A1 = B1 and A2 = B2, that is, A = B. □

Theorem 4.11

Let A, B ∈ ℂn,n. Then ACEB if and only if there exists a unitary matrix U^ satisfying

A=U^[TS^1S^200000N22]U^,(38a)

B=U^[TS^1T1S^1T1S^2T1S^1S10T1S100N2]U^,(38b)

where T and T1 are invertible, N22 and N2 are nilpotent, and N22N2.

Proof

Let ACEB, and A = A1 + A2 and B = B1 + B2 are the core-EP decompositions of A and B, respectively. Then A1#B1 and A2B2. By applying Lemma 2.5, Theorem 4.7 and A1#B1, we have

B1=U^[TS^1T1S^1T1S^2T1S^1S10T1S1000]U^,B2=U^[00000000N2]U^,

where U^, T, T1, [N11N12N21N22] and N2 are as in Theorem 4.7.

Since A2B2, we have rk (B2A2) = rk (B2) − rk (A2), that is,

rk([000N2][N11N12N21N22])=rk(N2)rk([N11N12N21N22]).(39)

In addition, it is easy to check that

rk(N2)rk([N11N12N21N22])rk(N2)rk(N22)rk(N2N22)rk([000N2][N11N12N21N22]).(40)

Applying (39) to (40) we obtain

rk(N22)=rk([N11N12N21N22])(41)

rk(N2)rk(N22)=rk(N2N22).(42)

Therefore, we obtain

N22N2.(43)

Since N22N2, there exist nonsingular matrices P and Q such that

N22=P[D100000000]Q,N2=P[D1000D20000]Q,

where D1 and D2 are nonsingular, (see [12, Theorem 3.7.3]). It follows that

rk(N22)=rk(D1)andrk(N2)rk(N22)=rk(D2).(44)

Denote

N12=[M12M13M14]QandN21=P[M21M31M41].(45)

Then

[N11N12N21N22]=[Irk(N11)00P][N11M12M13M14M21D100M31000M41000][Irk(N11)00Q]

and

rk([N11N12N21N22])=rk(D1)+rk([M13M14])+rk([M31M41])                          +rk(N11M12D11M21)

It follows from (44) and (41) that

M13=0,M14=0,M31=0andM41=0.(46)

Therefore,

[N11N12N21N2N22]=[Irk(N11)00P][N11M1200M2100000D200000][Irk(N11)00Q].

By applying (41), (44) and [N11N12N21N22][000N2], we derive that

rk([000N2][N11N12N21N22])=rk([N11M12M210])+rk(D2)=rk(N2)rk(N22)=rk(D2).

Therefore, [N11M12M210]=0, that is,N11 = 0,M12 = 0 and M21 = 0. By applying (45) and (46), we obtain N11 = 0, N12 = 0 and N21 = 0. So, we obtain (38a) and (38b).

Let A and B be of the forms as given in (38a) and (38b). It is easy to check that A = A1 + A2 and B = B1 + B2 are the core-EP decompositions of A and B, respectively, and

A1=U^[TS^1S^2000000]U^,A2=U^[00000000N22]U^;B1=U^[TS^1T1S^1T1S^2T1S^1S10T1S1000]U^,B2=U^[00000000N2]U^.

It follows from (23) and N22N2 that A1#B1 and A2B2. Therefore, ACEB. □

Remark 4.12

In Ex. 4.5, it is easy to check that A#,B. Since A1#B1, we have ACEB. Therefore, the corenilpotent partial order does not imply the CE partial order.

Corollary 4.13

Let A, B ∈ ℂn,n. If ACEB, then AB.

Proof

Let A, B ∈ ℂn,n. Then A and B have the forms as given in Theorem 4.11. According to ACEB, we have N22N2, that is,

rk(N2N22)=rk(N2)rk(N22).(47)

Since T and T1 are invertible, it follows that

rk(B)=rk(T)+rk(T1)+rk(N2);rk(A)=rk(T)+rk(N22);rk(BA)=rk([0T1S^1T1T1S^1S10T1S100N2N22])=rk([Irk(T)T1S^100Irk(T1)000Inrk(T)rk(T1)][0T1S^1T1T1S^1S10T1S100N2N22])=rk([T1S10N2N22])=rk([T100N2N22])=rk(T1)+rk(N2N22).(48)

Therefore, by applying (22), (47) and (48) we derive rk(BA) = rk(B) − rk(A), that is, AB.

5 Characterizations of the core-EP order

As is noted in [13], the core-EP order is given:

AB:A,BCn,n,AA=ABandAA=BA.(49)

Some characterizations of the core-EP order are given in [13].

Lemma 5.1 ([13])

Let A,B ∈ ℂn,n and AB. Then there exists a unitary matrix U such that

A=U[T1T2S10N11N120N21N22]U,B=U[T1T2S10T3S200N2]U,(50)

where [N11N12N21N22] and N2 are nilpotent, T1 and T3 are non-singular.

Theorem 5.2

Let A, B ∈ ℂn,n. Then AB if and only if

AAW=BAWandAAW=BAW.(51)

Proof

Let A be as given in (5), and denote

UBU=[B1B2B3B4].(52)

By applying (20a) and

BAW=UB1B2B3B4T1T2S00U=UB1T1B1T2SB3T1B3T2SU,

we have AAW=BAW if and only if

B1=TandB3=0.

It follows that

AAW=UT0SNT1T2S00U=UTT1TT2SST1ST2SU,BAW=UT0B2B4T1T2S00U=UTT1TT2SB2T1B2T2SU.

Therefore, AAW=BAW and AAW=BAW if and only if

B1=T,B3=0,B2=S,andB4isarbitrary,(53)

that is, A and B satisfy AAW=BAW and AAW=BAW if and only if there exists a unitary matrix U such that

A=U[TS0N]U,B=U[TS0B4]U,(54)

where N is nilpotent,T is non-singular and B4 is arbitrary. Therefore, by applying Lemma 5.1, we derive the characterization (51) of the core-EP order. □

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported partially by Guangxi Natural Science Foundation [grant number 2018GXNSFAA138181], China Postdoctoral Science Foundation [grant number 2015M581690], High level innovation teams and distinguished scholars in Guangxi Universities, Special Fund for Bagui Scholars of Guangxi [grant number 2016A17] and National Natural Science Foundation of China [grant number 11771076].

Acknowledgement:

The authors wish to extend their sincere gratitude to the referees for their precious comments and suggestions.

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About the article

Received: 2017-12-13

Accepted: 2018-09-12

Published Online: 2018-10-31


Since AA is core invertible, we use the symbol 3c in (7).


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1218–1232, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0100.

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© 2018 Wang and Chen, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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