Two Kinds of Inverse Problems for Hermitian Anti-reflexive Matrices
Liang Jun-ping · College Mathematics · 2009
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.