The reverse order law for \(\{1, 3, 4\}\)-inverse of the product of two matrices
type: Item
instance of: scholarly article
title: The reverse order law for \(\{1, 3, 4\}\)-inverse of the product of two matrices
zbMATH Open document ID: 1187.15005
author: Deqiang Liu
author: Hu Yang
published in: Applied Mathematics and Computation
publication date: 2010-03-31T00:00:00Z
review text: For an \(m \times n\) matrix \(A\) a generalized inverse of \(A\) is an \(n \times m\) matrix \(X\) which satisfies some of the following four Penrose equations: (1) \(AXA=A\), (2)\(XAX=X\), (3) \((AX)^*=AX\), (4) \((XA)^*=XA\). Let \(\emptyset\neq \eta\subseteq \{1, 2, 3, 4\}\). Then \(A\eta\) denotes the set of all matrices \(X\) which satisfy (i) for all \(i \in \eta\). Any matrix \(X \in A\eta\) is called an \(\eta\)-inverse of \(A\). The reverse order law for the generalized inverses has been studied by many mathematicians since the middle 1960s. In this paper, the authors investigate the relationship between \(\{1, 3,4\}\)-inverses of \(AB\) and the product of \(\{1, 3, 4\}\)-inverses of \(A\) and \(B\). They give some necessary and sufficient conditions for \(B\{1, 3, 4\}A\{1, 3, 4\} \subseteq (AB)\{1, 3, 4\}\), \(B\{1, 3, 4\}A\{1, 3, 4\} \supseteq (AB)\{1, 3, 4\}\) and \(B\{1, 3, 4\}A\{1, 3, 4\}=(AB)\{1, 3, 4\}\) and also present two numerical examples.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5688908
zbMATH Keywords: reverse order law
zbMATH Keywords: generalized inverse
zbMATH Keywords: maximal ranks
zbMATH Keywords: minimal ranks
zbMATH Keywords: Penrose equations
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: j.amc.2009.12.056
OpenAlex ID: W2053191399
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DOI: 10.1016/J.AMC.2009.12.056
description: scientific article; zbMATH DE number 5688908