The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices
type: Item
instance of: scholarly article
title: The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices
zbMATH Open document ID: 1028.15012
DOI: 10.1016/S0024-3795(02)00550-5
author: Fuzhao Zhou
author: Xiyan Hu
author: Lei Zhang
published in: Linear Algebra and its Applications
publication date: 2003-05-04T00:00:00Z
review text: A matrix \((a_{ij})\in \mathbb{R}^{n\times n}\) is called centro-symmetric if \(a_{ij}=a_{n+1-i,n+1-j}\). The following problem is solved in this paper. Given \(X\in \mathbb{R}^{n\times m}\), \(\Lambda=\text{diag}(\lambda_1,\dots,\lambda_{m}) \in \mathbb{R}^{m\times m}\). Find (all) centro-symmetric matrices \(A\) such that \(AX=X\Lambda\). A necessary and sufficient condition for the existence of such an \(A\) is obtained and set \(S_E\) of all solutions is described. Moreover for \(\tilde A\in \mathbb{R}^{n\times n}\) an expression for the unique \(A^\ast\in S_{E}\) for which \(\|\tilde A-A^\ast\|= \inf_{A\in S_{E}}\|\tilde A-A\|\) is written (here \(\|B\|=\text{tr}(B^{\top}B)\) for any \(B\in \mathbb{R}^{n\times n}\)).
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 65F18
zbMATH DE Number: 1903583
zbMATH Keywords: Centro-symmetric matrix
zbMATH Keywords: Inverse eigenvalue problem
zbMATH Keywords: Optimal approximation
reviewed by: Michal Zajac
MaRDI profile type: Publication
cites work: Designing the Hopfield neural network via pole assignment
cites work: The inverse problem for the vibrating beam
cites work: Inverse eigenvalue problem in structural design
cites work: A survey of matrix inverse eigenvalue problems
cites work: The numerically stable reconstruction of a Jacobi matrix from spectral data
cites work: Q3721690
cites work: Q3804078
cites work: Computing a nearest symmetric positive semidefinite matrix
cites work: Q4241092
cites work: Q5866924
cites work: Least squares solutions for a class of inverse problems for antisymmetric matrices
cites work: Least-squares solution of \(F=PG\) over positive semidefinite symmetric \(P\)
cites work: The Inverse of a Finite Toeplitz Matrix
cites work: The pseudoinverse of a centrosymmetric matrix
cites work: Q3359644
full work available at URL: s0024-3795(02)00550-5
OpenAlex ID: W2080003912
description: scientific article; zbMATH DE number 1903583