一类反三角矩阵的群逆和Drazin逆

The Group Inverses and Drazin Inverses of a Class of Anti-triangular Block Matrices

对于反三角矩阵M=(PQI0)的群逆和Drazin逆的研究,总是在块矩阵满足不同条件下进行的。本文在新的条件下获得了一些结论,即:当子块矩阵P可逆或ind(Q)≤1时,研究了M存在群逆的充要条件及群逆的表达式。同时根据这些结论,得到了当ind(P)≤1,PπQP=0和ind(P#QPP#)≤1时M的Drazin逆表达式,以及当PQQπ=0,Q2 QD+QπPQπ可逆时M的Drazin逆表达式。

The research on the group inverse and Drazin inverse of the anti-triangular block matrix M=（7was always done under some conditions for the sub-matrices. In this paper, some results areobtained under some new conditions, namely, when P is nonsingular or ind ind（Q）≤1, sufficient andnecessary conditions for the existence of the group inverse of M are developed and the expressions of thegroup inverse of M are presented. Moreover, based on the above results, the representations for Drazininverses of M are obtained under conditions ind（P）≤1,PπQP=0andind（P#QPP#）≤1, or on the conditions PQQπ=0,Q2 QD＋QπPQπ are nonsingular.