某些分块矩阵的Drazin逆和立方幂零矩阵

Drazin Inverse of Some Special Partitioned Matrices and Cubic Nilpontent Matrix

S.L.Campbell在文献[1]中提出的形如[A B C 0]的分块矩阵的Drazin逆的表达式问题至今没有完全得到解决。本文对如下特殊情形的2×2分块矩阵[A AA* AA* 0],[AA* A A 0],其中A为立方幂零矩阵,A*为A的共轭转置矩阵,利用Drazin逆和Moore-Penrose逆的关系及立方幂零矩阵性质,给出了这些分块矩阵的Drazin逆的表达式。

Campbell discussed the problem of expressing the generalized inverse of a partitioned block matrix[A B C 0],but the problem has not yet solved satisfactorily by scientists and researchers.Given the following partitioned block matrices [A AA＊ AA＊ 0],[AA＊ A A 0], where A is a cubic nilpontent matrix and A＊ is a transposed conjugate matrix of A,the expressions of Drazin inverses of those partitioned block matrices are given,by using formulas of Drazin inverses and Moore-Penrose inverses,and properties of cubic nilpontent matrix.