Moore-Penrose inverse positivity of interval matrices
type: Item
instance of: scholarly article
title: Moore-Penrose inverse positivity of interval matrices
zbMATH Open document ID: 1236.15012
author: M. Rajesh Kannan
author: K. C. Sivakumar
published in: Linear Algebra and its Applications
publication date: 2011-12-02T00:00:00Z
review text: The authors study (entrywise) nonnegativity of the Moore-Penrose inverse \(C^\dag\) of a matrix \(C\in\mathbb{R}^{m\times n}\). In particular they consider the following questions: (1) When is \(C^\dag\) nonnegative? (2) When is \(C^\dag\) nonnegative if \(C\leq B\), where \(B\in \mathbb{R}^{m\times n}\) is a given matrix with \(B^\dag\geq 0\)? (3) When is \(C^\dag\) nonnegative for all matrices \(C\in[A, B]\), where \(A^\dag\geq 0\) and \(B^\dag\geq 0\)? Each of these three questions is answered by necessary and sufficient conditions which are similar to known ones for regular \(n\times n\) matrices \(A\), \(B\), \(C\) in the same connection. One of these known criteria comes from Ortega and Rheinboldt and uses weak regular splittings, another one goes back to Krasnosel'skij, Lifshits, and Sobolev and uses some part of the range of \(C\), and a third one was published by Rohn and answers question (3) for regular matrices. For the analogue of the latter one the concept of a range kernel regular interval matrix \([A,B]\) is introduced by \(R(A)= R(B)\) and \(N(A)= N(B)\), where \(R(\cdot)\) denotes the range and \(N(\cdot)\) the kernel of an \(m\times n\) matrix. For such an interval matrix a particular subset \(K\) is introduced by virtue of which various additional results could be derived.
reviewed by: Günter Mayer
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 65G30
zbMATH DE Number: 5985512
zbMATH Keywords: interval matrix
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: range kernel regularity
zbMATH Keywords: proper splitting
zbMATH Keywords: weak regular splitting
zbMATH Keywords: inverse positive matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2011.07.012
OpenAlex ID: W2083307776
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DOI: 10.1016/J.LAA.2011.07.012
description: scientific article; zbMATH DE number 5985512