广义正定矩阵的特殊乘积

SPECLAL PROOUCTS OF POSITIVE DEFINITE MATRICES

设H_n={A|A∈C^(n×n),A~*=A,且对所有的0≠x∈C^n,(x,Ax)=x~*Ax>0}。C_n={A|A∈C^(h×n),且对所有0≠x∈C^n,(x,Ax)= x~*Ax>0}。本文证明了下面事实:如果A∈H_n,B∈G_n,那么A(?)B,B(?)A和A·B∈G_n,同时我们有反例来说明如果A,B∈G_n,那么A(?)B,A·B∈G_n是不正确的。

Let H_n= {A|A∈C^(n×n),, A = A and (x, Ax)=x Ax>o for all 0≠x∈C^n} G_n = {A|A∈C^(n×n), and (x, Ax) =x Ax>0 for all 0≠x∈C^n}. It is shown that if A∈H_n, B∈G_n, then A×B. B×A and A·B∈G_n. We also findcounterexamples to show that it is not true that if A, B∈G_n, then A×B, A·B∈G_n.