The Existence And The Representations Of The Group Inverse For Some Partitioned Matrices

Let Km×n denote the set of all the m×n matrices over a skew field K.For A∈Kn×n, the smallest nonnegative integer k such that rank(Ak)=rank(Ak+1) is called the index of A, denoted by Ind(A). For A∈Kn×n, the matrix X∈Kn×n satisfying the following equations AkXA=Ak, XAX=X, AX=XA is called the Drazin inverse of A, denoted by AD, where Ind(A)≤k. When Ind(A)≤1, X is called the group inverse of A, we denoted X by A#. A#exists iff rank (A2)=rank (A). if A#exists, it is unique.The group inverse of the matrix has numerous applications. For example, the difference for the identity matrix and the transition matrix of Markov chain is group invertible, and the group inverse play an important role in the biological, financial and engineering. The group inverse has also many applications in singular differential equations, linear equations and differential equations. Moreover, the group inverse has been applied to the research areas of iteration method and cryptography.In1979, Campbell and Meyer proposed an open problem to find the representation for the Drazin inverse of the block matrix(?)(A,Dare square). In1983, Campbell proposed another open problem to find the representation of the Drazin inverse for the block matrix when study the solution of second 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, Many scholars have given some expressions for the Drazin inverse and the group inverse of block matrix(?)under some special conditions.In Chapter1, 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. The main results of this paper are given in Chapter3and4which are listed below: 1.Let the block matrix M=(?)where m,n are positive integers, if the matrix M satisfies one of the following conditions, then the existence and the representations of M#are given:(i) rank(B)≥rank(C);(ii) CA=CB.We also the representations of under some conditions.where m, n are positive integers, if rank(B)≤rank(C) and AB=BA. then the existence and the representations of the group inverse for M are given. We also give the representations of(?)under some conditions.3.let the block matrix M=(?), if A, C are group inverse AC=BC and AπB=0, then the representations of M are given. We also give the representations of under some conditions.4.1et the block matrix M=(?), if A, C are group inverse AC=-BC and AπB=0, then the representations of the group inverse for M are given. We also give the representations of(?)under some conditions.