Further results on the group inverses and Drazin inverses of anti-triangular block matrices

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Abstract

In this paper, the sufficient and necessary conditions for the existence of the group inverse for a class of 2 × 2 anti-triangular block matrices M with some certain conditions satisfied, and representations of the group inverse of M are obtained, which can be regarded as the extension of Bu et al. [C. Bu, J. Zhao, J. Zheng, Group inverse for a class 2 × 2 block matrices over skew fields, Appl. Math. Comput. 204 (2008) 45–49] and [C. Bu, J. Zhao, K. Zhang, Some results on group inverses of block matrices over skew fields, Electron. J. Linear Algebra, 18 (2009) 117–125]. Further, we use these results to determine the expressions of the Drazin inverse of M. Finally, two numerical examples are given to illustrate our results.

Introduction

Let Cm×n be the set of complex m × n matrices. The symbols r(A) and A(1) will denote the rank and any {1}-inverse of a given matrix ACm×n. Further, AD will stand for the Drazin inverse of A, i.e., the unique matrix satisfying the equations [1]AkXA=Ak,XAX=X,AX=XA,where k = Ind(A), the index of A, is the smallest nonnegative integer k such thatr(Ak+1)=r(Ak).

If Ind(A)=1, then the Drazin inverse of A is reduced to the group inverse, denote by A#. If Ind(A)=0, then AD=A-1. In addition, we denote Aπ=I-AAD (or Aπ=I-AA#), especially, if A is idempotent, then Aπ=I-A.

The Drazin inverse is very useful, and has various applications in singular differential or difference equations, Markov chains, cryptography, iterative method and numerical analysis. In 1979, Campbell and Meyer first proposed an open problem of finding a formula for the Drazin inverse of M=ABCD in terms of the blocks of the partition over complex fields, where A and D are square matrices but need not to be the same size. To the best of our knowledge, up to now, this problem has not been solved completely. However, some special cases of the Drazin inverse and group inverse of M were studied under certain conditions [3], [7], [10], [11], [15], [17], [18], [19], [21], [23], and the expressions of the Drazin inverse and group inverse in terms of Banachiewicz–Schur forms were also investigated in [12], [13], [14], [16], [20].

Specially, when D=0,M=ABC0is the so-called anti-triangular block matrix, where ACn×n,BCn×m and CCm×n. In recent years, the Drazin inverse and group inverse for M formed by (1.2) have been considered by many authors. For instance, Deng [6] established some expressions of the Drazin inverse of M when BCAπ=0 and CADB is either invertible or equal to zero. Deng and Wei [9] got explicit expressions for the Drazin inverse of the anti-triangular block matrix M under the following different cases(i)ABC=0;(ii)BCAπ=0,(I-Aπ)BC=0;(iii)AπAB=0,BC(I-Aπ)=0.In [22], Castro–González and Dopazo provided the representations for MD under two different conditions(i)A=I,B=I;(ii)CAAD=C,ADBC=BCAD.

Bu et al. [2], [4] determined the conditions for the existence of M# in the case B=A or C=A, and presented the representations of M#. For the matrixM=AAC0,if A2=A, in [2], the authors showed that M# exists if and only if r(C)=r(CAC); for each matrix A, need not to be idempotent, if r(C)r(A), in [4], they showed that M# exists if and only if r(C)=r(A)=r(CA)=r(AC). In recent paper [8], Bu et al. considered the group inverse of M in the case A can be expressed as the linear combination or product combination of matrices B and C. Cao and Li [25] mainly considered the group inverse of matrix with the form AX+YBAB0 over skew fields. Zhou et al. [26] established the conditions for the existence and the representations for the group inverse of block matrix M with one or two full rank sub-blocks.

In the next section, some improved results on the group inverse of the anti-triangular block matrix M are obtained, which extend the results given by Bu et al. [2], [4]. Moreover, applying our new results, several representations of the Drazin inverse of anti-triangular block matrix M are derived under certain conditions.

Before giving the main results, we first introduce some important lemmas as follows.

Lemma 1.1 [24]

Let ACm×n,BCm×k,CCl×n and DCl×k. ThenrAB=r(A)+r((I-AA(1))B)=r((I-BB(1))A)+r(B),rAC=r(A)+r(C(I-A(1)A))=r(A(I-C(1)C))+r(C),rABCD=r(A)+r0(I-AA(1))BC(I-A(1)A)D-CA(1)B,where the A(1) is an arbitrary {1}-inverses of A.

Lemma 1.2

Let BCm×n,CCn×m. If r(B)=r(BCB), then (BC)# and (CB)# exist. Furthermore, (BC)#=B((CB)#)2C.

Proof

In view of r(B)=r(BCB), we haver(B)=r(BCB)r(BC)r(B),which meansr(B)=r(BC).Similarly, we can deducer(B)=r(CB).Hencer(B)=r(BCB)=r(BC)=r(CB).And the condition r(B) = r(BC) shows that there exists a matrix X ϵ Cm×n such that B = BCX. Thereforer(BC)=r(BCB)=r(BCBCX)r(BC)2.On the other hand,r(BC)r(BC)2.Combining (1.6), (1.7) gives r(BC) = r(BC)2. Therefore, (BC)# exists.The existence of (CB)# can be proved similarly, we omit the details.

With the fact (BC)D=B((CB)D)2C, then (BC)#=B((CB)#)2C is evident. 

Lemma 1.3 [2]

Let M=0B0A, where ACm×m and BCn×m. Then M# exists if and only if A# exists and rBA=r(A).

Lemma 1.4 [5]

Let P,QCm×m, and Ind(P)=r,Ind(Q)=s, such that PQ=0. Then(P+Q)D=Qπi=0s-1Qi(PD)i+1+i=0r-1(QD)i+1PiPπ.Specially,

(i) If P2=0, then(P+Q)D=QD+(QD)2P.

(ii) If Q2=0, then(P+Q)D=PD+Q(PD)2.

Section snippets

Main results

In this section, we first establish the necessary and sufficient conditions for the existence of M# under some weaker conditions than those in [2], [4], where M is given by (1.2), then present the explicit expressions for M#. Furthermore, we use our new results to derive some representations for the Drazin inverse of M under certain assumptions.

First, we consider the case that A is idempotent.

Theorem 2.1

Let M be partitioned as (1.2). If A2=A and CA=C, then

(i) M# exists if and only if (CB)# exists and rAπB

Examples

In this section, we give two examples to illustrate Theorem 2.3 and Theorem 2.7.

Example 3.1

Consider the block matrix M=ABC0, whereA=101010000,B=1-111-11,C=111202.It is easy to verify that A2=A,CAπB=0 and (CB)# exists. Furthermore, we can calculate thatAπ=00-1000001,(CB)#=1100,(CB)π=0-101.Hence, applying Theorem 2.3, we getMD=-6-1-622-30-311717-2-2111-1-120200.

Example 3.2

Consider the block matrix M=ABC0, whereA=101021000,B=110100,C=110-111.We observe that R(A)R(B), and r(B)=r(BCB). Furthermore, we can compute that(BC

Acknowledgements

The authors would like to thank the editors and the reviewers for their valuable comments and helpful suggestions, which improved the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11171361) and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20110191110033).

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