两个幂等矩阵组合的群逆

Group Inverse of Combinations of Two Idempotent Matrices

运用幂等矩阵核空间的性质证明复数域上两个非零幂等矩阵P,Q的组合a1P＋b1Q＋a2PQ＋b2QP＋a3PQP＋b3QPQ＋a4PQPQ＋b4QPQP＋a5PQPQP＋b5QPQPQ＋a6PQPQPQ（其中ai,bj∈C（1≤i≤6,1≤j≤5）且a1b1≠0）在条件（PQ）3=（QP）3下的秩与系数的选取无关,进而证明其群逆的存在性,并得到了组合aP＋bQ＋cPQ＋dQP的群逆计算公式.

With the aid of the properties of the null space of idempotent matrices,the rank of the combination a1P ＋b1Q ＋a2PQ ＋b2QP ＋a3PQP ＋b3QPQ ＋a4PQPQ ＋b4QPQP ＋a5PQPQP ＋b5QPQPQ ＋a6PQPQPQ of two nonzero idempotent matrices P and Q over the complex field C,where ai,bj∈ C（1≤i≤6,1≤j≤5）and a1b1≠0,was proved to be independent with the choice of its coefficients and the group inverse of the combination was furtherly verified to exist under the condition（PQ）3=（QP）3.Moreover,the formula for the group inverse of the combination aP＋bQ＋cPQ＋dQP was also obtained.