The Solvability Conditions for Inverse Eigenvalue Problem of Anti-Bisymmetric Matrices

Abstract

This paper is mainly concerned with solving the following two problems:
Problem I. Given $X$ $\in$ $ C^{n \times m} $, $\Lambda = {\rm diag}( \lambda_1, \lambda_2, \dots, \lambda_m) \in C^{m\times m}$. Find $ A \in ABSR^{n \times n} $ such that $$AX=X\Lambda$$where $ABSR^{n \times n}$ is the set of all real $n\times n$ anti-bisymmetric matrices.

Problem Ⅱ. Given $A^* \in R^{n \times n}$. Find $\hat{A} \in S_E $ such that $$||A^* - \hat{A}||_F=\underset{A\in S_E}{\min}||A^*- A ||_F,$$where $||\cdot||_F$ 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.