Representations for the Drazin inverse of the sum P+Q+R+S and its applications

Let P, Q, R and S be complex square matrices and M=P+Q+R+S. A quadruple (P,Q,R,S) is called a pseudo-block decomposition of M ifPQ=QP=0PS=SQ=QR=RP=0andRD=SD=0,where RD and SD are the Drazin inverses of R and S, respectively. We investigate the problem of finding formulae for the Drazin inverse of M. The explicit representations for the Drazin inverses of M and P+Q+R are developed, under some assumptions. As its application, some representations are presented for 2×2 block matricesAB0CandABDC, where the blocks A and C are square matrices. Several results of this paper extend the well known representation for the Drazin inverse ofAB0Cgiven by Hartwig and Shoaf, Meyer and Rose in 1977. An illustrative example is given to verify our new representations.