一类反对称正交反对称矩阵逆特征值问题

A Class of Inverse Eigenvalue Problems for Anti-symmetric Ortho-symmetric Matrices

设P为一给定的对称正交矩阵,记AARnP={A∈Rn×n‖AT=-A,(PA)T=-PA}.讨论了下列问题:问题给定X∈Cn×m,Λ=diag(λ1,λ2,…,λm).求A∈AARPn使AX=XΛ.问题设A~∈Rn×n,求A*∈SE使‖A^-A*‖=infA∈SE‖A^-A‖,其中SE为问题的解集合,‖.‖表示Frobenius范数.研究了AARPn中元素的通式,给出了问题解的一般表达式,证明了问题存在唯一逼近解A*,且得到了此解的具体表达式.

Let P ∈ R^n×m such that P^T = P, P^-1 = P^T. Set AARГ^N = {A ∈ R^n×m ｜｜ A^T = A, （PA）^T = PA}. This paper discuss the following two problems： Problem Ⅰ. Given X E ∈^n×m, A = diag（λ1, λ2,…, λm,）. Find A ∈ AARГ^n such that AX = XA. Problem Ⅱ. Given ^～A ∈ R^n×m. Find A^＊ ∈ SE suchthat ｜｜^～A-A^＊｜｜ = A∈SE inf ｜｜^～A- A｜｜ , where SE is the solution set of Problem Ⅰ, ｜｜·｜｜ is the Frobenius norm. In this paper, the general expression of the elements in AARP^n is studied, and the sufficient and necessary conditions under which SE is nonempty are obtained. The general form of SE has been given. The expression of the solution A^＊ of Problem Ⅱ is presented.