Volume 20, Issue 3
The Solvability Conditions for Inverse Eigenvalue Problem of Anti-Bisymmetric Matrices

J. Comp. Math., 20 (2002), pp. 245-256.

Published online: 2002-06

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• 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.

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@Article{JCM-20-245, author = {Xie , Dong-XiuHu , Xi-Yan and Zhang , Lei}, title = {The Solvability Conditions for Inverse Eigenvalue Problem of Anti-Bisymmetric Matrices}, journal = {Journal of Computational Mathematics}, year = {2002}, volume = {20}, number = {3}, pages = {245--256}, 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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8914.html} }
TY - JOUR T1 - The Solvability Conditions for Inverse Eigenvalue Problem of Anti-Bisymmetric Matrices AU - Xie , Dong-Xiu AU - Hu , Xi-Yan AU - Zhang , Lei JO - Journal of Computational Mathematics VL - 3 SP - 245 EP - 256 PY - 2002 DA - 2002/06 SN - 20 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8914.html KW - Eigenvalue problem, Norm, Approximate solution. AB -

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.

Dong-Xiu Xie, Xi-Yan Hu & Lei Zhang. (1970). The Solvability Conditions for Inverse Eigenvalue Problem of Anti-Bisymmetric Matrices. Journal of Computational Mathematics. 20 (3). 245-256. doi:
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