Minimization problems for \((R,S)\)-symmetric and \((R,S)\)-skew symmetric matrices
type: Item
instance of: scholarly article
title: Minimization problems for \((R,S)\)-symmetric and \((R,S)\)-skew symmetric matrices
zbMATH Open document ID: 1059.15019
author: William F. Trench
published in: Linear Algebra and its Applications
publication date: 2004-11-22T00:00:00Z
review text: A matrix \(A\in {\mathbb C}^{m\times n}\) is \((R,S)\)-symmetric (resp. \((R,S)\)-skew symmetric) if \(RAS=A\) (resp. \(RAS=-A\)). The class of all such matrices \(A\) is denoted by \({\mathcal S}\). Here \(R\) and \(S\) are non-trivial involutions, i.e. \(R=R^{-1}\neq \pm I\), \(S=S^{-1}\neq \pm I\). Let \(Z\in {\mathbb C}^{n\times q}\), \(W\in {\mathbb C}^{m\times q}\). The author studies the following problems: {(i)} Give necessary and sufficient conditions for the existence of \(A\in {\mathcal S}\) such that \(AZ=W\), and find all such matrices \(A\). {(ii)} Find \(\sigma (Z,W)=\min _{A\in {\mathcal S}}\| AZ-W\| \), and characterize the class \({\mathcal S}(Z,W)=\{ A\in {\mathcal S} \mid \| AZ-W\| =\sigma (Z,W)\}\). {(iii)} If \(B\in {\mathbb C}^{m\times n}\) is arbitrary, find \(\sigma (Z,W,B)=\min _{A\in {\mathcal S}(Z,W)}\| A-B\| \), and find \(A\in {\mathcal S}(Z,W)\) which realizes this minimum. The author obtains explicit formulas for all matrices from \(\sigma (Z,W)\) and \(\sigma (Z,W,B)\).
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A29
zbMATH DE Number: 2117279
zbMATH Keywords: approximation
zbMATH Keywords: Frobenius norm
zbMATH Keywords: involution
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: \((R,S)\)-symmetric matrix
zbMATH Keywords: \((R,S)\)-skew symmetric matrix
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: j.laa.2004.03.035
OpenAlex ID: W1982947218
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DOI: 10.1016/J.LAA.2004.03.035
description: scientific article; zbMATH DE number 2117279