| For A∈Cn×n, the smallest nonnegative integer k such that r(Ak)=r(Ak+1) is called the index of A, denoted by Ind(A). For A∈Cn×n, the unique matrix X∈Cn×n satisfying the following equations AkXA=Ak, XAX=X, AX=XA is called the Drazin inverse of A, denoted by AD, where k=Ind(A).The Drazin inverses of block matrices have important applications in singular differential equations, Markov chains, the problem of least squares, numerical analysis and so on. A lot of scholars do the related research. In1977, C. D. Meyer gave the formula for the Drazin inverse of the upper triangular matrix firstly. In1979, Campbell and C. D. Meyer proposed an open problem to find the representation for the Drazin inverse of the block matrix (?).But the problem have been not solved until now. In1983, Campbell proposed an open problem to find an explicit representation for the Drazin inverse of the block matrix (?) in the background of the singular differential system. However, The problems have been not solved completely. In recent years, Some scholars gave some expressions for the Drazin inverse of the2×2block matrix (?) under some certain conditions.The Chapter1and2of this paper briefly introduce the development status, the research significance, the research situation and the basic knowledge of the generalized inverses. The Chapter3gives the representations for the Drazin inverse of M=(?)(where A and D are square matrices) with the generalized Schur complement S=D-CADB nonsingular and the blocks satisfying the one of the following conditions:(Ⅰ) BCAπ=0, BDCAπ=0, D2CAπ=0;(Ⅱ) BCAπB=0, DCAπB=0, DCAπA=0; (Ⅲ)AπBC=0,AπBDC=0,AπBD2=0;(Ⅳ)CAπA2=0,CAπBC=0,CAπBD=0,CAπAB=0;(Ⅴ)AπBD=0,AπBCA=0,BCAπB=0.In Chapter4, this paper obtains the representations for the Drazin inverse of M=(?)(where A and D are square matrices)witll the blocks satisfying the one of the following conditions:(1)BCAπB=0,AπBCA=0and S=D-CADB=0;(2)ABCAπ=0,AπBCAπ=0and S=D-CADB=0;(3)ABCA=0,CBCA=0,A2BC=0,CABC=0and S:D-CADB=0;(4)CB=0,A2B=0,CAB=0. |
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