Structure of a nonnegative regular matrix and its generalized inverses
type: Item
instance of: scholarly article
title: Structure of a nonnegative regular matrix and its generalized inverses
zbMATH Open document ID: 0885.15015
DOI: 10.1016/S0024-3795(97)00062-1
published in: Linear Algebra and its Applications
publication date: 1998-04-20T00:00:00Z
review text: A matrix \(A\) is nonnegative (positive) if each entry of the matrix is nonnegative (positive). A nonnegative matrix \(A\) is called regular if it admits a nonnegative generalized inverse. In this paper, a complete description of all nonnegative generalized inverses of a nonnegative regular matrix \(A\) is given. In particular, the author proves the following important result: Let \(A\) be an \(m\times n\) regular matrix of rank \(r\) with no zero row or column, and let \(G\) be a nonnegative generalized inverse of \(A\). Then \(G\) is dominated by \(A'\), \(A'\) is the transpose of \(A\). Furthermore, there exist permutation matrices \(P\), \(Q\) such that \[ PAQ=\begin{bmatrix} \begin{array}{cccc|c} A_{11} & 0 & \cdots & 0 & {} \\ 0 & A_{22} & \cdots & 0 & \ast \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_{rr} & {} \\ \hline & & \ast & & \ast \end{array} \end{bmatrix}, \,Q^\prime GP^\prime= \begin{bmatrix} \begin{array}{cccc|c} G_{11} & 0 & \cdots & 0 & {} \\ 0 & G_{22} & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & G_{rr} & {} \\ \hline & & 0 & & \ast \end{array} \end{bmatrix} \] where \(A_{ii}\) is a positive, rank-one matrix and \(A_{ii}G_{ii} A_{ii}= A_{ii}\), \(i=1,2,\dots, r\).
reviewed by: Jule Zhang
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1100485
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: generalized inverse
zbMATH Keywords: regular matrix
author: R. B. Bapat
MaRDI profile type: Publication
cites work: Inverses of nonnegative matrices
cites work: Eight types of matrix monotonicity
cites work: Regular Nonnegative Matrices
cites work: The Generalized Inverse of a Nonnegative Matrix
cites work: Q5556138
cites work: A geometric treatment of non-negative generalized inverses
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description: scientific article; zbMATH DE number 1100485