The best generalized inverse
type: Item
instance of: scholarly article
title: The best generalized inverse
zbMATH Open document ID: 0983.15006
author: Gisèle Ruiz Goldstein
author: Jerome A. Goldstein
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-04-19T00:00:00Z
review text: If \(A\) is an \(m\times n\) matrix, an approximate solution to the problem \(Ax=y\), is \(x=By\), where \(B\) is an \(n\times m\) matrix, and it is called a generalized inverse of \(A\). The Moore-Penrose generalized inverse of \(A\) (\(\widetilde{A}=(|A|_{\mid\text{range}(|A|)})^{-1}U^{\ast }\), where \(|A|=(A^{\ast }A)^{1/2}\) is the absolute value of \(A\) and \(U\) is the linear and isometric map from \(|A|x\) to \(Ax\), as a function from \(\text{range}(|A|)\) to \(\text{range}(A)\)) minimizes \(\|Ax-y\|^{2}=\|(AB-I)y\|^{2}\) where \(\|\cdot \|\) is the Euclidean norm\(.\) The main result of the paper is that for all norms type ``Schatten-von Neumann'' the Moore-Penrose generalized inverse \(\widetilde{A}\) is best, that is, for any \(p\), \(1\leq p\leq \infty \), and for any \(n\times m\) matrix \(B\), \(\|A\widetilde{A}-I\|_{p}\leq \|AB-I\|_{p}\). This result is also established to the more general context of all unitarily invariant norms \(|||\cdot |||\), that is, \(\min\{|||AB-I|||\mid B\in \mathfrak{M}_{n\times m}\}=|||A\widetilde{A}-I|||\) for every unitarily invariant norm (a norm \(|||\cdot |||\) is called unitarily invariant if \(|||UAV|||\) \(=|||A|||\) for all \(A\in \mathfrak{M}_{m\times n}\) and all unitary operators \(U\), \(V\) in \( \mathfrak{M}_{m\times m}\), \(\mathfrak{M}_{n\times n}\), respectively).
reviewed by: M. Ladra
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1563450
zbMATH Keywords: generalized inverse
zbMATH Keywords: Moore-Penrose generalized inverse
zbMATH Keywords: Schatten-von Neumann norms
zbMATH Keywords: invariant norms
zbMATH Keywords: singular value decomposition
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6927
OpenAlex ID: W2072131484
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DOI: 10.1006/JMAA.2000.6927
description: scientific article; zbMATH DE number 1563450