In Addition To The Ring Some 2 X 2 Piece Of Group Inverse Matrix

The generalized inverse of matrix is an extremely active feld in the matrix the-ory, which has numerous applications in many areas, such as diferential equations,Markov chains, numerical analysis, cryptography, control theory and so on. As weknow, group inverse is a special case of Drazin inverse, but it doesn’t exist for ev-ery square matrix. Therefore, it is very necessary to study the existence and therepresentations of the group inverse.In1979, Campbell and Meyer proposed an open problem, that is to fnd the e x-plicit representation for the Drazin （group） inverse of the block matrix A B,C Dwhere A and D are square matrices. Until now, this problem has not been solvedcompletely. However, un der some conditio ns, there have been some results aboutthis problem. Let M=AX, where A, Y∈Km×n, X, B∈Kn^×m.In this paper, we give the necessary and sufcient conditions to the existence andthe representations of the group inverse for M under some conditions.This paper is organized as follows: In Chapter1, we introduce the researchstatus of the generalized inverse and the group inverse of matrices in domestic andoverseas. Also, the motivation and the main results of this paper are presented.In Chapter2, we study the group inverses of products of two matrices and givesome lemmas. In Chapter3, we give the necessary and sufcient conditions to theexistence and the representations of M when A, B, X, Y satisfy one of the followingconditions, which generalize the relative results:（1） A, B, X, Y∈K^×n, XA=AX and X is invertible, A exists;（2） Y=0, A∈Km×n, X, B∈Kn^×mand rank（B）≥rank（A）;（3） Y=0and replace X with XB, A∈Km×n, B∈Kn^×m, X∈K^×n.