Abstract:

Given P∈OR\+\{n×n satisfying  P\+T=P. Set  SAR\+n\-P={A∈R\+\{n×n|A\+T=A, (PA)\+T=-PA}.
This paper discusses the following problems
Problem Ⅰ〓Given X∈R\+\{n×m, Λ=diag(λ\-1,λ\-2, …, λ\-m)∈R\+\{m×m. Find A∈SAR\+n\-P such that AX=XΛ.Problem ⅡGiven B, X∈R\+\{n×m.Find A∈SAR\+n\-P such that ‖AX-B‖=min.Problem Ⅲ〓Given [AKA～]∈R\+\{n×n. FindA\+*∈S\-E such that‖[AKA～]-A\+*‖=inf[DD(X]A∈S\-E[DD)]‖[AKA～]-A‖ where S\-E is the solution set of Problem Ⅱ, ‖\5‖ is the Frobenius norm.The necessary and sufficient conditions for the solvability of  Problem Ⅰ have been studied. The general form of the solution set S\-E of Problem Ⅱ has been given. For Problem Ⅲ the expression of the solution A\+* has been provided.