Generalized inverses of symmetric \(M\)-matrices
type: Item
instance of: scholarly article
title: Generalized inverses of symmetric \(M\)-matrices
zbMATH Open document ID: 1195.15004
published in: Linear Algebra and its Applications
publication date: 2010-04-06T00:00:00Z
review text: The authors carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric \(M\)-matrices. The key idea in our approach is to identify any symmetric \(M\)-matrix with a positive semi-definite Schrödinger operator on a connected network whose conductances are given by the off-diagonal elements of the \(M\)-matrix. Moreover, the potential of the operator is determined by the positive eigenvector of the \(M\)-matrix. The authors prove that any generalized inverse can be obtained throughout a Green kernel plus some projection operators related to the positive eigenfunction. Moreover, the authors use the discrete Potential Theory associated with any positive semi-definite Schrödinger operator to get an explicit expression for any generalized inverse, in terms of equilibrium measures. Finally, the authors particularize the obtained result to the cases of tridiagonal matrices and circulant matrices.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 5689457
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: generalized inverses
zbMATH Keywords: equilibrium measure
zbMATH Keywords: triangular matrices
zbMATH Keywords: circulant matrices
zbMATH Keywords: singular irreducible symmetric \(M\)-matrices
zbMATH Keywords: positive eigenvector
zbMATH Keywords: Green kernel
zbMATH Keywords: positive eigenfunction
zbMATH Keywords: discrete Potential Theory
author: Enrique Bendito
author: A. Carmona
author: A. M. Encinas
author: M. Mitjana
MaRDI profile type: Publication
full work available at URL: j.laa.2009.11.008
OpenAlex ID: W1976719908
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DOI: 10.1016/J.LAA.2009.11.008
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description: scientific article; zbMATH DE number 5689457