The Solvability Conditions for Inverse Eigenvalue Problem of Anti-Bisymmetric Matrices (Journal of Computational Mathematics, 2002)
This paper is mainly concerned with solving the following two problems: Problem I. Given $X$ $\in$ $ C^{n × m} $, $Λ = {\rm diag}( λ_1, λ_2, …, λ_m) \in C^{m× m}$. Find $ A \in ABSR^{n × n} $ such that $$AX=XΛ$$where $ABSR^{n × n}$ is the set of all real $n× n$ anti-bisymmetric matrices.Problem Ⅱ. Given $A^* \in R^{n × n}$. Find $\hat{A} \in S_E $ such that $$||A^* - \hat{A}||_F=\underset{A\in S_E}{\min}||A^*- A ||_F,$$where $||·||_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of Problem Ⅰ have been studied. The general form of $ S_E $ has been given. For Problem Ⅱ the expression of the solution has been provided.
Keywords: Eigenvalue problem; Norm; Approximate solution.