Characterization of generalized inverses by a rank equation.
type: Item
instance of: scholarly article
title: Characterization of generalized inverses by a rank equation.
zbMATH Open document ID: 1054.15006
DOI: 10.1016/S0096-3003(03)00322-9
author: Bing Zheng
author: R. B. Bapat
published in: Applied Mathematics and Computation
publication date: 2004-03-29T00:00:00Z
review text: Consider the \(2 \times 2\) block-rank-equation rank\((A_{11},A_{12};A_{21},X) = \) rank\((A_{11})\) for blocks \(A_{11},A_{12}\), \(A_{21}\) and \(X\) of suitable sizes. If \(A_{11} \) is nonsingular then it is well-known that \(A_{11}^{-1}\) is the unique matrix \(X\) satisfying the above equation where \(A_{12}=A_{21}=I\). Similar results are known for the Moore-Penrose inverse and the group inverse of \(A_{11}\). In this paper, any outer inverse of \(A_{11} \in {\mathbb C}^{m \times n}\) is characterized by a similar rank equation (setting \(A_{12} = A_{11}X\) and \(A_{21}=XA_{11}\)). Moreover, the converse problem is solved for reflexive inverse and Drazin inverses. In fact, using the singular value (the core-nilpotent, resp.) decomposition, all matrices \(A_{12}\) and \(A_{21}\) such that a reflexive inverse (the Drazin inverse, resp.) of \(A_{11}\) is a solution of the above equation are found.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 2062684
zbMATH Keywords: Rank equation
zbMATH Keywords: outer inverse
zbMATH Keywords: core-nilpotent decomposition
zbMATH Keywords: singular value decomposition
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: group inverse
zbMATH Keywords: reflexive inverse
zbMATH Keywords: Drazin inverses
reviewed by: Néstor Thome
MaRDI profile type: Publication
cites work: Q4061081
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cites work: A characterization and representation of the generalized inverse \(A_{T,S}^{(2)}\) and its applications
cites work: Q5181471
cites work: Linear Algebra and Linear Models
cites work: Representation and approximation of the outer inverse \(A_{T,S}^{(2)}\) of a matrix \(A\)
cites work: Generalized inverses and a block-rank equation.
full work available at URL: s0096-3003(03)00322-9
OpenAlex ID: W2021358927
description: scientific article; zbMATH DE number 2062684