Hermitian非自反矩阵的两类反问题

记J为一广义反射矩阵,HAJn×n为关于J的n阶Hermitian非自反矩阵的集合.本文考虑如下两个问题:问题Ⅰ给定X,B∈n×m,求A∈HAJn×n,使得‖AX-B‖=min.问题Ⅱ给定X∈n×m,B∈n×n,求A∈HAJn×n,使得XHAX=B.首先利用奇异值分解讨论问题Ⅰ的解的通式,然后利用广义奇异值分解得到了问题Ⅱ有解的充分必要条件和解的通式,最后给出问题Ⅰ和Ⅱ的逼近解的具体表达式.

Two Kinds of Inverse Problems for Hermitian Anti-reflexive Matrices

Given a generalized reflection matrix J let HAJn×n denote the set of Hermitian anti-reflexive matrices of order n on J:HAJn×n={A|A=AH,JAJ=-A,A∈∈Cn×n}.The following three problems are considered in this paper.Problem Ⅰ: Given X,B∈Cn×m,find A∈HAJn×n, such that ‖AX-B‖=min.Problem Ⅱ: Given X∈Cn×m,B∈Cn×n,find A∈HAJn×n,such that XHAX=B.For Problem Ⅰ,a general expression of the least-square solution is obtained by using the singular value decomposition method.For Problem Ⅱ,the necessary and sufficient conditions and a general expression solution are discussed by using the generalized singular value decomposition method.At last,we derive the representation of the unique optimal approximations from the solutions of problems Ⅰ and Ⅱ.