任意域上解约束矩阵方程的Cramer法则

摘要

Let Fm×n be the set of all m × n matrices over a field F. For a matrix A ∈ Fm×n, we denote the column space and right kernel space of A by R(A) and Nr(A) , respectively. If m = n, we also denote by Ind(A) the index of A. A Cramer rule for finding the unique solution of a class of restricted matrix equationsWAWX (～W)B (～W) = D , R( X) ()R( ( AW) k1 ) , Nr( X) () Nr( ( (～W)B )(~k)2 )is presented, whereA ∈Fm×n , W∈Fn×m, B∈Fp×q, W∈Fq×p, D∈Fn×p , R(D)()R((WA)k2), Nr(D)()Nr( (B (～W))(~k)1 ), k1= Ind(AW), k2= Ind( WA), (~k)1 = Ind( B (～W)) and (~k)2 = Ind((～W)B). This generalizes those results in [ 15,16,17 ] from the field of complex numbers to any field.%Fm×n表示域F上所有m×n矩阵的集合.R(A)和Nr(A)分别表示矩阵A∈Fm×n的列空间和核空间.若m=n,用Ind(A)定义矩阵A的指标.给出了求一类约束矩阵方程WAWX (～W)B (～W)=D, R( X) ()R( (AW)k1 ), Nr( X) ()Nr( ((～W)B)(~k)2 )的唯一解的Cramer法则,其中A∈Fm×n,W∈Fn×m,B∈Fp×q,(～W)∈Fq×p,D∈Fn×p,R(D)()R((WA)k2),Nr(D)()Nr((B(～W)(~k)1),k1=Ind(AW),k2=Ind(WA),(~k)1=Ind(B(～W)),(~k)2=Ind((～W)B).这将[15-17]中的结果从复数域推广到任意域.