体上一类具有幂等子块的分块矩阵的群逆

Group inverse of a class of block matrices with idempotent sub-blocks over skew fields

设K是一个体,Km×n表示m×n上所有K矩阵的集合.对矩阵A∈K若存在矩阵X∈Kn×n使AXA=A,XAX=X,AX=XA,则称X为A的群逆.研究分块矩阵广义逆的表达式是矩阵广义逆理论中研究的重要问题.分块矩阵的群逆表达式在奇异微分和差分方程、马尔可夫链、迭代方法和密码学等领域有广泛应用.这里给出了体上分块矩阵[A B/B0](A,B∈Kn×n,B2=B,((I-B)A)#存在)的群逆的存在性及表示形式.

Suppose K is a skew field and K is the set of all the n x n matrices over K. For A ∈K^m×n if there exists a matrix X∈K^m×n satisfying the matrix equations AXA =A ,XAX =X,AX =XA, then we call X the group inverse of A. Research on representations of the generalized inverses is an important problem in the theory of generalized inverses of matrices. The group inverse of block matrices has important applications in singular differential and difference equations, Markov chains, iterative methods, cryptography and so on. The existence and the representa- ~ 2 tion of the block matrix （A,B ∈K^m×n ,B^2 = B, （ （I - B） A） wexists） are given in this paper.