Representations Of Drazin Inverses For Some Block Matrices

Drazin inverses of the matrices have important applications in many fields, and it is an important content in generalized inverse of matrices theory. In 1979, C.D. Meyer proposed an pen problem to find the representation for the Drazin inverse of the block matrix(?). In 1983, Campbell proposed an open problem to find an explicit representation of the Drazin inverse for the form of (?) when he researched the solution of differential equations. However, as the limitations of the methods and the difficultly of the problem, these problems have been not completely solved. In recent years, some expressions for the Drazin inverse and the group inverse of block matrix (?) have been given under some certain conditions.Let C be the complex number field, and Cm×n be the set of all m×n matrices over C Denote the rank of A by rank (A). For a matrix A∈Cm×n, if the matrix X∈Cm×n satisfies the matrix equations: AkXA=Ak, XAX=X, AX=XA, then X is called the Drazin inverse of A, and is denoted by X=AD, where k is the smallest nonnegative integer such that rank(Ak+1)= rank(Ak) and is called the index of A. Denote the index of A by Ind(A). In the case that Ind(A)≤1, the matrix X is called the group inverse of A and is denoted by X=A#.In Chapter 1, this paper briefly gives the development status and the research significance of generalized inverses of the matrices, and also gives the basic knowledge of the generalized inverses theory of the matrices in Chapter2. Finally, the main results of this paper are given in Chapter3 and 4, which are listed below:1. Let M=(?), where A and Dare square matrices, the generalized Schur complement of A in M is S=D-CADB=0 and the matrix M satisfies one of the following conditions:(1) BCAπis r-nilpotent matrix and (I+BC(AD)2)ABCAπ=0;(2) CAπBC=0 and AAπBC=0;(3) CAπBCAπ=0, (A+BCAD)BCAπ=0;(4) AπBC is p-nilpotent matrix and (A+ADBC)AπBC=0;(5) ADBC=0 and AAπB=0, the representations of Drazin inverse for M are given.2.The necessary and sufficient conditions for the existence and exact expression of the group inverse for M=(?) are given over a skew field.3.A new proof of the representation for the Drazin inverse of the matrix M=(?) is given under the condition of DA=0,DB=0 or CD=0,BD=0.4.A new proof of the representation for the Drazin inverse of the matrix M=(?) given under the condition of BC=0,BDC and BD2=0.