When a matrix and its inverse are nonnegative
type: Item
instance of: scholarly article
title: When a matrix and its inverse are nonnegative
zbMATH Open document ID: 1339.15022
published in: Missouri Journal of Mathematical Sciences
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.mjms/1404997113
reviewed by: Ali Reza Moghaddamfar
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 6359510
zbMATH Keywords: stochastic matrix
zbMATH Keywords: permutation matrix
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: canonical form
author: J. Ding
author: N. H. Rhee
MaRDI profile type: Publication
cites work: Q3631986
cites work: Q4485695
cites work: Q5342712
review text: A matrix is called stochastic if it is a nonnegative matrix for which each of its row sums equals \(1\). It is clear that if \(A\) is a permutation matrix, then \(A\) and \(A^{-1}\) are stochastic. The converse of this statement is also true and its proof has been given by the authors in [``Teaching tip: when a matrix and its inverse are stochastic'', Coll. Math. J. 44, No. 2, 108--109 (2013; \url{doi:10.4169/college.math.j.44.2.108})].NEWLINENEWLINE In this article, they present another proof of this fact based on the canonical forms of stochastic matrices. They also extend this result to show that that \(A\) and \(A^{-1}\) are nonnegative if and only if \(A\) is a product of a diagonal matrix with all positive diagonal entries and a permutation matrix.
description: scientific article; zbMATH DE number 6359510