The least-squares solutions of inconsistent matrix equation over symmetric and antipersymmetric matrices
type: Item
instance of: scholarly article
title: The least-squares solutions of inconsistent matrix equation over symmetric and antipersymmetric matrices
zbMATH Open document ID: 1047.65026
DOI: 10.1016/S0893-9659(03)00041-7
author: Xiyan Hu
author: Dongxiu Xie
author: Yan-Ping Sheng
published in: Applied Mathematics Letters
publication date: 2004-06-11T00:00:00Z
review text: Let \(SA^n\) denote the set of jointly symmetric and antipersymmetric matrices \(A\in{\mathbb R}^{n\times n}\) whose elements, by definition, satisfy the pair of formulae \(a_{ij}=a_{ji}\) and \(a_{ij}=-a_{n-j+1,n-i+1}\) for \(i,j=1,2,\ldots,n\). The authors use singular value decomposition to find explicitly the set \({\mathcal S}\subset SA^n\) of general solutions \(L\) which minimize \(\| LG-E\| \) (Problem 1) in the Frobenius norm for \(G,E\in{\mathbb R}^{n\times n}\). Next, given \(L^*\in{\mathbb R}^{n\times n}\), the unique solution \(\widehat L\in {\mathcal S}\) which minimizes \(\| L^*-\widehat L\| \) (Problem 2) in the Frobenius norm is also obtained by means of the singular value decomposition. In the course of the proofs, expressions for matrices in \(SA^{2k}\) and \(SA^{2k+1}\) are obtained in terms of explicit orthogonal similarities. Algorithms for matrices of even and odd orders are applied to numerical examples to illustrate the theory. In particular, as \(L^*\) approaches a solution to Problem 1, the norm \(\| L^*-\widehat L\| \) becomes smaller. References 8 to 19 are not quoted in the paper.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 2073769
zbMATH Keywords: matrix equation
zbMATH Keywords: least-squares solution
zbMATH Keywords: optimal approximate
zbMATH Keywords: matrix norm
zbMATH Keywords: singular value decomposition
zbMATH Keywords: algorithms
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2073769