The Representations Of The Group Inverses For Some 2×2 Block Matrices

The generalized inverses of block matrices have inclusive applications in numerical analysis, Markov chains, linear differential equation, difference equation and so on. The research on the representations of generalized inverses of the block matrices is a very important problem in matrice theory, and the representations of the Drazin and group inverse of matrices are the most extensive. This paper mainly studies the existence and the representations of the group inverses of a class of block matrices.Let K be a skew field, and Km×n be the set of all m×n matrices over K. For A∈Kn×m, if there exists the matrix X∈Kn×n satisfying the following matrix equations: AXA=A,XAX=X,AX=XA we said the matrix X is the group inverse of A, and is denoted by X=A#. When the matrix A is nonsingular, A#=A-1.This paper mainly introduce the development overview, the research significance and the research status of generalized inverse of the matrices in Chapter1,and the basic knowledge of generalized inverses of matrices in Chapter 2. The main results in this paper are given in chapter 3,4 and 5, which are listed below:(1) The existence of the group inverse of the matrix M=(?)(A is square and A2=A) is given, by the formula of M#= M(M3)(1)M, the representations of the group inverse of the matrix M is also given in Chapter 3;(2) The existence and representations of the group inverse of block matrix M=(?), (A, B, C∈{A, A#, AA#}) are given in Chapter 4;(3) An equivalent definition of the group inverse is given with the rank relation of the matrix in Chapter 5, by the definition, the representation of the group inverse of the block matri M=(?) where A,B,C,D∈Kn×n, A2=A.