On some matrix equalities for generalized inverses with applications
type: Item
instance of: scholarly article
title: On some matrix equalities for generalized inverses with applications
zbMATH Open document ID: 1165.15006
author: Yongge Tian
author: George P. H. Styan
published in: Linear Algebra and its Applications
publication date: 2009-05-12T00:00:00Z
review text: A matrix \(X\) is called a \(g\)-inverse of \(A\), denoted by \(X = A^-\), if it satisfies \(AXA = A\). The collection of all possible \(g\)-inverses of \(A\) is denoted by \(\{A^-\}\). The authors obtain some necessary and sufficient conditions for the matrix equality \(A^- = PN^-Q\) to hold, where \(A \in \mathbb{C}^{m\times n}\), \(N \in \mathbb{C}^{k\times l}\), \(P \in \mathbb{C}^{n\times l}, Q \in \mathbb{C}^{k\times m}\), see also \textit{J. Gross} and \textit{Y. Tian} [Linear Algebra Appl. 417, No. 1, 94--107 (2006; Zbl 1105.15005)]. In particular, they give identifying conditions for the set equalities \(\{C^-\}=\{A^- + B^-\}\) and \(\{A^-\} =\{A_1^- + \cdots+ A_k^-\}\) to hold.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5552970
zbMATH Keywords: generalized inverse
zbMATH Keywords: matrix equality
zbMATH Keywords: additive decomposition
zbMATH Keywords: block decomposition
zbMATH Keywords: parallel sum of matrices
zbMATH Keywords: shorted matrix
zbMATH Keywords: Schur complement
zbMATH Keywords: matrix rank method
MaRDI profile type: Publication
full work available at URL: j.laa.2008.12.005
OpenAlex ID: W2003785292
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DOI: 10.1016/J.LAA.2008.12.005
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description: scientific article; zbMATH DE number 5552970