Weak group inverse

Hongxing Wang; Jianlong Chen

doi:10.1515/math-2018-0100

Open Access Published by De Gruyter Open Access October 31, 2018

Weak group inverse

Hongxing Wang and Jianlong Chen

From the journal Open Mathematics

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 ℂ_m,n is the set of m × n matrices with complex entries; A^*, ℛ(A) and rk(A) represent the conjugate transpose, range space (or column space) and rank of A ∈ ℂ_m,n, respectively. Let A ∈ ℂ_n,n be singular, the smallest positive integer k satisfying rk (A^k+1) = rk (A^k) 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 ℂnCM stands for a set of n × n matrices of index less than or equal to 1. The Moore-Penrose inverse of A ∈ ℂ_m,n is defined as the unique matrix X ∈ ℂ_n,m satisfying the equations:

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

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

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

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

AX=AA†,ℛ(X)⊆ℛ(A)

and is denoted as X=A◯# [2]. When A∈ℂn CM , 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 ∈ ℂ_n,n with Ind (A) = k. The DMP inverse of A is A^d,† = A^DAA^† [3]. The BT inverse of A is A^⋄ = (A²A^†)^† [4, Definition 1]. The core-EP inverse of A is A◯†=AkA∗kAk+1−A∗k [5, Theorem 3.5 and Remark 2]. It is evident that A◯#=A♢=Ad,†=A◯† in case of A∈ℂn CM . More results on the core inverse and related problems can be seen in [6–10].

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 ∈ ℂ_n,nwith 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 ∈ ℂ_n,nwith rk(A) = r. Then there exists a unitary matrix U such that

A=U[ΣKΣL00]U∗,(2)

where Σ = diag(σ₁I_r₁,σ₂I_r₂,. . ., σ_tI_r₁) is the diagonal matrix of singular values of A, σ₁ > σ₂ > … ≥ σ_t > 0, r₁ + r₂ + … + r_t = r, and K ∈ ℂ_r,r, L ∈ ℂ_r,n−rsatisfy KK^* + LL^* = I_r.

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

A◯#=UT−1000U∗,(3)

A#=U[T−1T−2S00]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 ∈ ℂ_n,nwith Ind(A) = k, then A can be written as the sum of matrices Â₁and Â₂, i.e. A = Â₁ + Â₂, where

A^1∈ℂn CM ,A^2k=0 and A^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 ∈ ℂ_n,nwith Ind(A) = k, then A can be written as the sum of matrices A₁and A₂, i.e. A = A₁ + A₂, where

(i) A1∈ℂn CM ;
(ii) A2k=0;
(iii) A1∗A2=A2A1=0.

Here one or both of A₁and A₂can be null.

Lemma 2.5 ([13])

Let the core-EP decomposition of A ∈ ℂ_n,nbe 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◯†=UT−1000U∗.(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 ∈ ℂ_n,n with Ind(A) = k, and consider the system of equations¹

2′ AX2=X,3c AX=A◯†A.(7)

Theorem 3.1

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

X=U[T−1T−2S00]U∗.(8)

Proof

Let A ∈ ℂ_n,n with Ind(A) = k. Since A◯†=AkA∗kAk+1−A∗k,RA◯†A⊆RA. Therefore, (3^c) is consistent. Let A be as in (5). From (6), we obtain

A◯†2A=UT−1T−2S00U∗(9)

and

AA◯†2A=A◯†A,(10)

that is, A◯†2A is a solution to (3^c).

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

AA◯†2A2=A◯†2A,(11)

that is, A◯†2A 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=A◯†AX=A◯†A◯†A=A◯†AY=AY2=Y,

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

Definition 3.2

Let A ∈ ℂ_n,nbe a matrix of index k. The WG inverse of A, denoted asA◯W, is defined to be the solution to the system (7).

Remark 3.3

WhenA∈ℂn CM , we haveA◯W=A#.

Remark 3.4

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

Remark 3.5

Let A ∈ ℂ_n,n. Applying Theorem 3.1, it is easy to checkA◯WAA◯W=A◯WandRA◯W=RAk.

More details about the weak Drazin inverse can be seen in [14–16].

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

Example 3.6

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

A†=[0.500001−100.50000010] and AD=[1011010100000000],

the DMP inverse A^d,† and the BT inverse A^⋄are

Ad,†=ADAA†=[1010010000000000] and A⋄=(A2A†)†=[0.500001000.50000000],

and the core-EP inverseA◯†and the WG inverseA◯Ware

A◯†=1000010000000000and A◯W=1010010100000000.

3.2 Characterizations of the WG inverse

Theorem 3.7

Let A ∈ ℂ_n,nbe as in (5). Then

A◯W=A1#=UT−1T−2S00U∗.(12)

Proof

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

Theorem 3.8

Let A ∈ ℂ_n,nwith Ind(A) = k. Then

A◯W=AA◯†A#=A◯†2A=A2◯†A.(13)

Proof

Let A be as in (5). Then

AA◯†A=UTS0NT−1000TS0NU∗=UTS00U∗,A◯†2=UT−1000U∗2=UT−2000U∗,A2◯†=UT2TS+SN0N2U∗◯†=UT−2000U∗.

It follows from Theorem 3.7 that

AA◯†A#=UTS00U∗#=UT−1T−2S00U∗=A◯W,A◯†2A=A2◯†A=UT−2000TS0NU∗=UT−1T−2S00U∗=A◯W.

Therefore, we obtain (13). □

Theorem 3.9

Let A ∈ ℂ_n,nwith Ind(A) = k. Then

A◯W=AkAk+2◯#A=A2PAk†A.(14)

Proof

Let A be as in (5). Then

Ak=U[TkΦ00]U∗,(15)

where Φ=∑i=1kTi−1SNk−i. It follows that

AkAk+2◯#A=UTkΦ00T−(k+2)000TS0NU∗=UT−1T−2S00U∗=A◯W,(16)

PAk=AkAk†=UIrk(Ak)000U∗,A2PAk†A=UT2000†TS0NU∗=A◯W.(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 ∈ ℂ_n,nwith Ind(A) = k. Then

rkA◯W=rkAD=rkAk.

It is well known that (A²)^D = (A^D)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

A2◯W=UT−2T−4TS+SN00U∗,A◯W2=UT−2T−3S00U∗.(19)

Therefore, A2◯W=A◯W2 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 ∈ ℂ_n,nbe as in (5). ThenA2◯W=A◯W2if 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

AA◯W=UTS0NT−1T−2S00U∗=UIT−1S00U∗;(20a)

A◯WA=UT−1T−2S00TS0NU∗=UIT−1S+T−2SN00U∗.(20b)

Therefore, we have the following Corollary 3.12.

Corollary 3.12

Let the core-EP decomposition of A ∈ ℂ_n,nbe as in (5). ThenAA◯W=A◯WAif and only if SN = 0.

Corollary 3.13

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

A◯W=AD=Ak+1◯#Ak=At+1◯†At,

where t is an arbitrary positive integer.

Proof

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

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

Ak−1=U[Tk−1Tk−2S0Nk−1]U∗, Ak=U[TkTk−1S00]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)S00TkTk−1S00U∗=UT−1T−2S00U∗=A◯W.

Therefore, A◯W=AD=Ak+1◯#Ak.

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

At=U[TtTt−1S0Nt]U∗, At+1=U[Tt+1TtS0Nt+1]U∗.

It follows from Lemma 2.5 that

At+1◯†=UT−(t+1)000U∗,At+1◯†At=UT−(t+1)000TtTt−1S0NtU∗=A◯W,(21)

Therefore, we derive A◯W=At+1◯†At, 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] :

A≤−B:A,B∈ℂm,n,rk(B−A)=rk(B)−rk(A),(22)

A≤#B:A,B∈ℂn CM ,A2=AB=BA,(23)

A≤DB:A,B∈ℂn,n,A^1≤#B^1,(24)

A≤#,−B:A,B∈ℂn,n,A^1≤#B^1 and A^2≤−B^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:

A≤ WG B:A,B∈ℂn,n, if A1≤#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 A≤ WG B and B≤ WG C, 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 A≤ WG C.

Example 4.1

Let

A=[111001000], B=[111002000]

AlthoughA≤ WG BandB≤ WG A, A ≠ B. 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 onℂn CM .

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 thatA≤ WG B.

Since A^DA ≠ A^D, we deriveA≰DB. Therefore, the WG order does not imply the Drazin order.

Example 4.5

Let

A^=[100000000],B^=[100001000],P=[1−20010001],A=PA^P−1=[120000000],B=PB^P−1=[12−2000000],A1=[120000000], A2=0, B1=[12−2000000], B2=[000001000],

in which A = A₁ + A₂and B = B₁ + B₂are the core-EP decompositions of A and B, respectively. ThenA≤DBandA1≰#B1. Therefore, the Drazin order does not imply the WG order.

It is well known that A≤DB⇒A2≤DB2, 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 deriveA2≰ WG B2. Therefore,A≤ WG B⇏A2≤ WG B2.

Theorem 4.7

Let A, B ∈ ℂ_n,n. ThenA≤ WG Bif and only if there exists a unitary matrixU^such that

A=U^[TS^1S^20N11N120N21N22]U^∗,(27a)

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

where T and T₁ are invertible, [N11N12N21N22]and N₂are nilpotent.

Proof

Assume A≤ WG B. 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

U∗B1U=[B11B12B21B22].(28)

Applying (12) gives

B1A1#=U[B11B12B21B22][T−1T−2S00]U∗=U[B11T−1B11T−2SB21T−1B21T−2S]U∗.

Since A≤ WG⁡B,A1≤#⁡B1. It follows from A1A1#=B1A1# that

B11=T and B21=0.(29)

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

A1#A1=U[IT−1S00]U∗,A1#B1=U[IT−1B12+T−2SB2200]U∗.

It follows from A1#A1=A1#B1 that

T−1(S−T−1SB22−B12)=0.

Therefore,

B12=S−T−1SB22,(30)

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

B1=U[TS−T−1SB220B22]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[TS−T−1SB220U1[T1S100]U1∗]U∗=U[I00U1][TSU1−T−1SU1U1∗B22U10[T1S100]][I00U1∗]U∗=U^[T[S^1S^2]−T−1[S^1S^2][T1S100]0[T1S100]]U^∗=U^[TS^1−T−1S^1T1S^2−T−1S^1S10T1S1000]U^∗.(34)

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

4.2 CE partial order

Consider the binary relation:

A≤ CE B:A,B∈ℂn,n,A1≤#B1 and A2≤−B2,(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 ∈ ℂ_n,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 writeA≤ CE B.

Remark 4.9

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

A≤ CE B⇒A≤ WG B.(36)

Furthermore,

A≤ CE B⇔A≤ WG B and A2≤−B2.(37)

Theorem 4.10

The CE order is a partial order.

Proof

Reflexivity is trivial.

Let A≤ CE B, B≤ CE C 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 A2≤−B2, B2≤−C2. Therefore A1≤#C1 and A2≤−C2. It follows from Definition 4.8 that A≤ CE C

If A≤ CE B and B≤ CE A, Then A₁ = B₁ and A₂ = B₂, that is, A = B. □

Theorem 4.11

Let A, B ∈ ℂ_n,n. ThenA≤ CE Bif and only if there exists a unitary matrixU^satisfying

A=U^[TS^1S^200000N22]U^∗,(38a)

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

where T and T₁are invertible, N₂₂and N₂are nilpotent, andN22≤−N2.

Proof

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

B1=U^[TS^1−T−1S^1T1S^2−T−1S^1S10T1S1000]U^∗,B2=U^[00000000N2]U^∗,

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

Since A2≤−B2, we have rk (B₂ −A₂) = 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(N2−N22)≤rk([000N2]−[N11N12N21N22]).(40)

Applying (39) to (40) we obtain

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

rk(N2)−rk(N22)=rk(N2−N22).(42)

Therefore, we obtain

N22≤−N2.(43)

Since N22≤−N2, 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) and rk(N2)−rk(N22)=rk(D2).(44)

Denote

N12=[M12M13M14]Q and N21=P[M21M31M41].(45)

Then

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

and

rk([N11N12N21N22])=rk(D1)+rk([M13M14])+rk([M31M41]) +rk(N11−M12D1−1M21)

It follows from (44) and (41) that

M13=0,M14=0, M31=0 and M41=0.(46)

Therefore,

[−N11−N12−N21N2−N22]=[Irk(N11)00P][−N11−M1200−M2100000D200000][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^1−T−1S^1T1S^2−T−1S^1S10T1S1000]U^∗,B2=U^[00000000N2]U^∗.

It follows from (23) and N22≤−N2 that A1≤#B1 and A2≤−B2. Therefore, A≤ CE B. □

Remark 4.12

In Ex. 4.5, it is easy to check thatA≤#,−B. SinceA1≰#B1, we haveA≰ CE B. Therefore, the corenilpotent partial order does not imply the CE partial order.

Corollary 4.13

Let A, B ∈ ℂ_n,n. IfA≤ CE B, thenA≤−B.

Proof

Let A, B ∈ ℂ_n,n. Then A and B have the forms as given in Theorem 4.11. According to A≤ CE B, we have N22≤−N2, that is,

rk(N2−N22)=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(B−A)=rk([0−T−1S^1T1−T−1S^1S10T1S100N2−N22])=rk([Irk(T)T−1S^100Irk(T1)000In−rk(T)−rk(T1)][0−T−1S^1T1−T−1S^1S10T1S100N2−N22])=rk([T1S10N2−N22])=rk([T100N2−N22])=rk(T1)+rk(N2−N22).(48)

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

5 Characterizations of the core-EP order

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

A≤◯†⁡B:A,B∈Cn,n,A◯†A=A◯†BandAA◯†=BA◯†.(49)

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

Lemma 5.1 ([13])

Let A,B ∈ ℂ_n,nandA≤◯†⁡B. Then there exists a unitary matrix U such that

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

where[N11N12N21N22]and N₂are nilpotent, T₁and T₃are non-singular.

Theorem 5.2

Let A, B ∈ ℂ_n,n. ThenA≤◯†⁡Bif and only if

AA◯W=BA◯WandA∗A◯W=B∗A◯W.(51)

Proof

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

U∗BU=[B1B2B3B4].(52)

By applying (20a) and

BA◯W=UB1B2B3B4T−1T−2S00U∗=UB1T−1B1T−2SB3T−1B3T−2SU∗,

we have AA◯W=BA◯W if and only if

B1=T and B3=0.

It follows that

A∗A◯W=UT∗0S∗N∗T−1T−2S00U∗=UT∗T−1T∗T−2SS∗T−1S∗T−2SU∗,B∗A◯W=UT∗0B2∗B4∗T−1T−2S00U∗=UT∗T−1T∗T−2SB2∗T−1B2∗T−2SU∗.

Therefore, AA◯W=BA◯W and A∗A◯W=B∗A◯W if and only if

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

that is, A and B satisfy AA◯W=BA◯W and A∗A◯W=B∗A◯W 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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1
Since A◯†A is core invertible, we use the symbol 3^c in (7).

Received: 2017-12-13

Accepted: 2018-09-12

Published Online: 2018-10-31

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Weak group inverse

Abstract

1 Introduction

2 Preliminaries

Lemma 2.1 ([1])

Lemma 2.2

Lemma 2.3

Lemma 2.4

Lemma 2.5 ([13])

3 WG inverse

3.1 Definition and properties of the WG inverse

Theorem 3.1

Proof

Definition 3.2

Remark 3.3

Remark 3.4

Remark 3.5

Example 3.6

3.2 Characterizations of the WG inverse

Theorem 3.7

Proof

Theorem 3.8

Proof

Theorem 3.9

Proof

Corollary 3.10

Corollary 3.11

Corollary 3.12

Corollary 3.13

Proof

4 Two orders

4.1 WG order

Example 4.1

Theorem 4.2

Remark 4.3

Example 4.4

Example 4.5

Example 4.6

Theorem 4.7

Proof

4.2 CE partial order

Definition 4.8

Remark 4.9

Theorem 4.10

Proof

Theorem 4.11

Proof

Remark 4.12

Corollary 4.13

Proof

5 Characterizations of the core-EP order

Lemma 5.1 ([13])

Theorem 5.2

Proof

Disclosure statement

Funding

Acknowledgement:

References

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