Group inverses of \(M\)-matrices associated with nonnegative matrices having few eigenvalues
type: Item
instance of: scholarly article
title: Group inverses of \(M\)-matrices associated with nonnegative matrices having few eigenvalues
zbMATH Open document ID: 0833.15004
DOI: 10.1016/0024-3795(94)00301-S
author: M. Neumann
author: S. J. Kirkland
published in: Linear Algebra and its Applications
publication date: 1995-07-03T00:00:00Z
review text: Several contributions to the study of the group generalized inverse \(A^\#\) of singular irreducible \(M\)-matrices \(A = rI - M\) are given in this note. The first results are explicit formulas for \(A^\#\), if \(M\) is of low rank. They allow to study when \(A^\#\) is an \(M\)-matrix, too. These results are applied to two interesting classes of matrices \(M\), namely doubly regular tournament matrices and magic squares of order \(4k\), as they are produced by MATLAB. Their eigenvalues are determined, and as their ranks are low, the group inverse can be explicitly given. Also their singular values are exhibited.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 769026
zbMATH Keywords: group generalized inverse
zbMATH Keywords: singular irreducible \(M\)-matrices
zbMATH Keywords: doubly regular tournament matrices
zbMATH Keywords: magic squares
zbMATH Keywords: MATLAB
zbMATH Keywords: eigenvalues
zbMATH Keywords: singular values
reviewed by: Ludwig Elsner
describes a project that uses: Matlab
MaRDI profile type: Publication
cites work: Construction of a set of test matrices
cites work: Q4787959
cites work: Q3943082
cites work: Nonnegative alternating circulants leading to \(M\)-matrix group inverses
cites work: Derivatives of the spectral radius as a function of non-negative matrix elements
cites work: Random evolutions and the spectral radius of a non-negative matrix
cites work: Shorter Notes: Convexity of the Dominant Eigenvalue of an Essentially Nonnegative Matrix
cites work: Algebraic multiplicity of the eigenvalues of a tournament matrix
cites work: Derivatives of the Perron root at an essentially nonnegative matrix and the group inverse of an M-matrix
cites work: On the first and second order derivatives of the Perron vector
cites work: On convexity properties of the spectral radius of nonnegative matrices
cites work: Convex spectral functions
cites work: Using the QR Factorization and Group Inversion to Compute, Differentiate, and Estimate the Sensitivity of Stationary Probabilities for Markov Chains
cites work: Q5591853
cites work: Taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices
cites work: Convexity and Concavity of the Perron Root and Vector of Leslie Matrices with Applications to a Population Model
cites work: The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains
cites work: The Condition of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities
cites work: Derivatives and Perturbations of Eigenvectors
cites work: A Remark on Characteristic Polynomials
cites work: Q4057472
cites work: Q5674306
Wikidata QID: Q127186405
description: scientific article; zbMATH DE number 769026