关于两个幂等矩阵组合群逆的探讨

Discussions on the Group Inverses of Combinations of Two Idempotent Matrices

运用矩阵零空间的性质证明了复数域上两个不同的非零幂等矩阵P,Q的组合a1P＋b1Q＋a2PQ＋b2QP＋…＋a（2n-1）（PQ）（n-1 ）P＋b（2n-1）（QP）（n-1 ）Q＋a（2n）（PQ）n（其中a1,b1,…,b（2n-1）,a（2n）∈C,a1,b1≠0）在条件（QP）n=0（n≥2）下的秩与系数的选取无关,进而证明了其群逆存在.另外,还得到了组合aP＋bQ＋cPQ＋dQP在条件（QP）n=0下的群逆表达式.

By using the properties of null space of matrices,the rank of the combinations a1P＋b1Q＋a2PQ＋b2QP＋…＋a（2n-1）（PQ）（n-1 ）P＋b（2n-1）（QP）（n-1 ）Q＋a（2n）（PQ）（n ）of two different nonzero idempotent matrices Pand Qover the complex field C,where a1,b1,…,a（2n）∈C,a1,b1≠0,was proved to be independent with the choice of its coefficients and under the condition（QP）n=0（n≥2）.Therefore,the existence of the group inverse of the combination was also obtained.In addition,the formula for the group inverse of the combination aP＋bQ＋cPQ＋dQP was presented under the condition（QP）n=0.