两个幂等矩阵的组合的群逆

ON THE GROUP INVERSE OF COMBINATIONS OF TWO IDEMPOTENT MATRICES

本文研究了当P与Q是两个复数域上的n阶幂等矩阵且满足PQP=PQ时,组合aP+bQ+cP Q+dQP+eQP Q的群逆问题,利用矩阵的分块及群逆的性质,证明了它是群逆阵,并且给出了其群逆的表达式,其中ab=0,a,b,c,d,e为复数.

This paper studies the group inverse problem of the combination aP ＋ bQ ＋ cP Q ＋ dQP ＋ eP QP, where P and Q are two idempotent matrices satisfying the condition P QP = P Q. By using the block decomposition of matrices and properties of group inverse, the combination is proved to be group invertible and the formulae of its group inverse is also obtained, where a, b, c, d, e are complex numbers with a, b nonzero.