两个幂等矩阵组合的Group逆

The Group Inverse of Combinations of Two Idempotent Matrices

讨论了复数域上两个非零的幂等矩阵P,Q的组合a1P＋b1Q＋a2PQ＋b2QP＋a3PQP＋b3QPQ＋a4PQPQ＋b4QPQP＋a5PQPQP＋b5QPQPQ＋a6PQPQPQ的group逆的存在性及表达式问题,其中ai,bj∈C（1≤i≤6,1≤j≤5）且a1b1≠0.运用幂等矩阵核空间的性质证明了该组合在条件（P Q）-3=（QP）-3下的秩与系数的选取无关并进而证明了其group逆存在.另外,还给出了组合aP＋bQ＋cPQ＋dQP的group逆计算公式.

The paper discusses the existence and the expressions of the combinations a1P＋b1Q＋a2PQ ＋b2QP＋a3PQP＋b3QPQ ＋a4PQPQ ＋b4QPQP＋a5PQPQP＋b5QPQPQ ＋a6PQPQPQ of two nonzero idempotent matrices Pand Qover the complex field C,where ai,bj∈C（1≤i≤6,1≤j≤5）and a1b1≠0.By the properties of the null space of idempotent matrices,the rank of the combination is proved to be independent with the choice of its coefficients and the existence of the group inverse of the combination is further verified under the condition（PQ）-3=（QP）-3. Moreover,the formular for the group inverse of the combinations aP＋bQ＋cPQ＋dQPis also presented.