Expression for the Group Inverse of the 2×2 Block Matrix

(Department of Mathematics,Southeast University,Nanjing211102,China;School of Mathematical and Physical Sciences,Xuzhou Institute of Technology,Xuzhou221006,China)

Expression for the Group Inverse of the 2×2 Block Matrix

DU Fa-peng

(Department of Mathematics,Southeast University,Nanjing211102,China;School of Mathematical and Physical Sciences,Xuzhou Institute of Technology,Xuzhou221006,China)

generalized inverse;group inverse;block matrix

§1.Introduction

Let C be a matrix algebra on complex feldBwith unitI.For a matrixA∈C,if there is a matrixB∈C such thatABA=A,BAB=B,thenBis called a generalized inverse ofA, denoted byA+.The matrixBis called the Drazin inverse ofA,ifBsatisfed

Bis denoted byAD.The least integerkis called the index ofA,denoted by ind(A).The idempotent matrixI-AADis denoted byAπ.If ind(A)=1,then we callBthe group inverse ofA,denoted byA#.In this caseAπ=I-AA#.

§2.Preliminaries

In this section,we give some important lemmas.

Lemma 2.1[25,30]LetA∈C.Then the following conditions are equivalent

(1)A#exists;

(2)AA++A+A-Iis invertible in C for someA+;

(3)A2A++I-AA+is invertible in C for someA+;

(4)A2A++I-AA+is invertible in C for anyA+;

(5)A+I-AA+is invertible in C for someA+;

(6)A+I-AA+is invertible in C for anyA+.

Lemma 2.2[23]LetA,B∈C andPbe a nonzero idempotent matrix in C.PutX=PAP+PB(I-P).

(1)IfPAPis group invertible and(PAP)(PAP)#B(I-P)=PB(I-P),thenXis group invertible too andX#=(PAP)#+[(PAP)#]2PB(I-P).

(2)IfXis group invertible,then so is thePAPand(PAP)(PAP)#B(I-P)=PB(I-P).

The following proposition comes from[25].Here,we give another proof.

Lemma 2.3[25]LetA∈C such thatA#exists.Then

ProofLetW=A2A++1-AA+,thenWis invertible by Lemma 2.1 sinceA#exists. SetP=AA+,Y=PW.It is easy to check thatY#exists andY#=W-1P.Note that

Y Y#A(I-P)=A(I-P)andAA+W=WAA+=A2A+.Thus,by Lemma 2.2,we have

Proposition 2.4LetA∈C andTbe an idempotent matrix such thatTA=A.IfI+AT-Tis invertible,thenA#exists and

ProofPutW=I+AT-T;B=W-2A.Noting thatTW=WT=AT,we have

This indicateBis the group inverse ofA.

Proposition 2.5LetA,P,Q∈C withP,Qare invertible.LetB=PAQ.ThenB#exists ifAQP+I-AA+is invertible and

ProofNoting thatB+=Q-1A+P-1,we have

Thus,the assertion follows from Lemma 2.3.

LetX,Y∈C.For convenience,we setEX=I-XX+,FX=I-X+X.The following lemmas play an important role in the context which come from[25].

Lemma 2.6[25]LetX,Y∈C andU=EXY,Z=I+X+Y,S=X+XZFU.Then (X+Y)#exists ifX+Y+FUFSis invertible.In this case,

Lemma 2.7[25]LetX,Y∈C andV=Y FX,Z=I+Y X+,S=EVZXX+.Then (X+Y)#exists ifX+Y+ESEVis invertible.In this case,

In[3,Theorem 2.1],J Ben´ıtez,X J Liu and T P Zhu consider(X+Y)#whenX#,Y#exist andXY=0.Now,we give a simpler proof in virtue of Lemma 2.6.

Corollary 2.8[3,Theorem2.1]LetX,Y∈C withX#,Y#exist.IfXY=0,then (X+Y)#exists and

ProofSinceX#,Y#exist andXY=0,we haveX#Y=XY#=X#Y#=0 andXYπ=X,XπYπ=Yπ-XX#.Thus,U=Y,Z=I,S=XX#andFU=Yπ,FS=Xπ.

Noting that(I+Y Y#XX#)-1=I-Y Y#XX#,we have

By Lemma 2.6,we have

§3.Expression for the Group Inverse of 2×2 Block Matrix

has been obtained in[16,Theorem 2.2]and[22,Lemma 2.2]under some conditions,respectively. But there remain another problem need to be solved,that is,how to calculate(CB)#?In the following theorem,we get a better result.

Thus,M#exists ifEBFC-BCis invertible by Lemma 2.1.

Put Δ=(EBFC-BC)-1.

Noting thatEBΔ-1=EBFC=Δ-1FC,ΔBCC+=-C+,B+BCΔ=-B+.We have ΔEB=FCΔ,FC=ΔEBFC,CΔEB=0,FCΔB=0,BCΔ=-BB+,ΔBC=-C+Cand

Thus,

Thus,by Lemma 2.3,we have

Using the result in Theorem 3.1,we give explicit expression of(CB)#and(BC)#.

Proposition 3.2LetB,C∈C.Then(CB)#,(BC)#exist ifEBFC-BCis invertible. Put(EBFC-BC)-1=Δ,then

Using Theorem 3.1 and the equation(M2)#=(M#)2,we get

Here,Δ=(EBFC-BC)-1.

(1)(M+)11=A++(FAV+Z+A+C)(QBA+-(I-QZ)U+EA)-FAV+BA+;

(2)(M+)12=FAV+(I-ZQ)-A+CQ;

(3)(M+)21=(I-QZ)U+EA-QBA+;

(4)(M+)22=Q=FUS+EV.

From[27],we have

So,

Thus,by Lemma 2.1,M#exists ifthe matrixM+I-MM+is invertible andM#=(M+IMM+)-2Mby Lemma 2.3.

Noting that the expression ofM#is very complicated and the calculation is routine,we only discuss the necessary and sufcient conditions such thatM#exists for some special cases in the following.

(1)IfA#exists andU=AπC=0,then

Hence,M#exists if

is invertible.

(2)IfAis invertible,thenU=0=V,S=Zand

Hence,M#exists if

is invertible.

In addition,ifZ=0,thenM#exists ifI+BA-2Cis invertible.This is the result of Robert in[29].

(3)IfCis invertible,then

Thus,M#exists if

is invertible.

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O151.21

:A

1002–0462(2016)04–0412–10

Received date:2014-11-30

Foundation item:Supported by the Fund for Postdoctoral of China(2015M581688);Supported by the National Natural Science Foundation of China(11371089);Supported by the Specialized Research Fund for the Doctoral Program of Higher Education(20120092110020);Supported by the Natural Science Foundation of Jiangsu Province(BK20141327);Supported by the Foundation of Xuzhou Institute of Technology(XKY2014207)

Biography:DU Fa-peng(1974-),male,native of Xuzhou,Jiangsu,an associate professor of Xuzhou Institute of Technology,Ph.D.,engages in functional analysis application and operator algebra.

2000 MR Subject Classifcation:15A09,65F20