On the structure of stochastic matrices with a subdominant eigenvalue near 1
type: Item
instance of: scholarly article
title: On the structure of stochastic matrices with a subdominant eigenvalue near 1
zbMATH Open document ID: 0892.15017
DOI: 10.1016/S0024-3795(97)00333-9
author: Darald J. Hartfiel
author: Carl D. Meyer
published in: Linear Algebra and its Applications
publication date: 1998-03-31T00:00:00Z
review text: Let \(P\) be an \(n \times n\) stochastic matrix. Define the uncoupling measure \(\sigma(P)\) of \(P\) to be the minimum of \[ \Sigma_{i \in M_1, j \notin M_2} p_{ij} + \Sigma_{i \in M_2, j \notin M_1} p_{ij} \] over pairs \(M_1\) and \(M_2\) of disjoint subsets of \(\{1, \ldots, n\}\). The authors explore the relationship between \(\lambda_2(P)\), the second largest eigenvalue of \(P\) (in absolute value) and \(\sigma(P)\). They prove the easy result that if \(\lambda_2(P)\) is close to \(1\), then \(\sigma(P)\) must be small. The rest of the paper is devoted to examining the relationship between these quantities in specific examples.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1116070
zbMATH Keywords: stochastic matrices
zbMATH Keywords: subdominant eigenvalue
zbMATH Keywords: uncoupling
MaRDI profile type: Publication
cites work: Q5834413
cites work: Bounds for eigenvalues of doubly stochastic matrices
cites work: Bounds on distances between eigenvalues
cites work: Uniform Stability of Markov Chains
cites work: Q5805016
cites work: Sensitivity of the Stationary Distribution of a Markov Chain
cites work: Q3696432
full work available at URL: s0024-3795(97)00333-9
OpenAlex ID: W2020010559
Wikidata QID: Q127975072
description: scientific article; zbMATH DE number 1116070