体上某些分块矩阵的Drazin逆

令Ωn×n记体Ω上的所有n×n矩阵的集合.对于一个固定的A∈Ωn×n,若正整数k=min{l|Al+1X=Al对某个X∈Ωn×n},则称k为A的指标.如果X∈Ωn×n满足下面的方程组AX=XA,X2A=X,Ak+1X=Ak,其中k为A的指标,则称X为A的Drazin逆,当k=1时,A#=AD被称为A的群逆.Ωn×n的某些分块矩阵的Drazin逆和群逆的存在性和表示被给出.

Drazin inverses for some partitoned matrices over skew fields

Let Ωn×n be the set of all n × n matrices over a skew field Ω. The positive integer k is called the index of the matrix A in Ωn ×n if k = min{l|At+1X = At for some X ∈Ωn×n}. The matrix X is called the Drazin inverse of A if X satisfies the equations: AX=XA, X2A=X and Ak+1X=Ak. Denote A#=ADwhen k==1, and A# is called the group inverse of A. The existence conditions and representations of Drazin inverses and group inverses for some partitioned matrices in Ωn×n are given.