关于矩阵群逆的逆序律

得到了体上两个n阶方阵A,B的群逆A#,B#若存在,则其乘积的群逆(AB) #也存在,且(AB) #=B#A#成立的充分与必要条件是:存在n阶可逆矩阵P使得A =Pdiag(A1,A2 ,…,As) P- 1,B =Pdiag(B1,B2 ,…,Bs) P- 1且对于任意i(i=1 ,2 ,…,s)有Ai,Bi阶数相同,Ai,Bi为可逆矩阵或为0矩阵;又对i≠1有Ai Bi=0 .

Reverse order Law on Group Inverses of Matrices

This paper draws a conclusion that if the group inverse A# and B# of matrixes A and B with the rank n exist in the field, their product′s group inverse (AB)# also exists. Furthermore, the sufficient and necessary condition of (AB)#=B#A# is that there is the rank n reversible matrix P to make A=Pdiag(A_1, A_2, ..., A_s)P -1, and B=Pdiag(B_1, B_2, ..., B_s)P -1, and make A_i and B_i for arbitrary i(i=1, 2, ..., s) with the same rank. A_i and B_i are reversible matrix or zero matrix. If i≠1, the equation A_iB_i=0 may be set.