On explicit forms for ergodicity coefficients
type: Item
instance of: scholarly article
title: On explicit forms for ergodicity coefficients
zbMATH Open document ID: 0792.15009
DOI: 10.1016/0024-3795(93)90113-3
author: Adolf Rhodius
published in: Linear Algebra and its Applications
publication date: 1994-03-27T00:00:00Z
review text: Suppose \(\| \cdot \|\) is any vector norm on row vectors constituting \(\mathbb{R}^ n\), and \(S_ n\) \((n \geq 2)\) the set of all \(n \times n\) stochastic matrices. The definition of an ergodicity coefficient \(\tau\) for \(P=\{p_{ij}\}\), \(P \in S_ n\) with respect to the norm \(\| \cdot \|\) is \(\tau(P)=\sup \{\| {\underset \sim x}^ T P \|: {\underset \sim x}^ T \in \mathbb{H}^ n\), \(\| {\underset \sim x}^ T \|=1\}\) where \(\mathbb{H}^ n=\{ {\underset \sim x}^ T: {\underset \sim x}^ T \in \mathbb{R}^ n,\;{\underset \sim x}^ T {\underset \sim 1}_ n=0\}\). Explicit forms for \(\tau\) in terms of the entries of \(P\) are known for the \(\ell_ p\) norms when \(p=1\), \(p=\infty\). The central idea in this paper is to regard \(\| \cdot \|\) as a norm on the vector subspace \(\mathbb{H}^ n\) of \(\mathbb{R}^ n\) and \(P\) as an operator on \(\mathbb{H}^ n\). The general procedure provides further ergodicity coefficients, their explicit forms, and new eigenvalue bounds which are compared with the Deutsch-Zenger numerical radius. The author's bounds have been obtained by a different approach in a sequel to his paper by the reviewer [ibid. 191, 245-252 (1993; Zbl 0785.15012)]. This sequel appeared slightly earlier through an editorial oversight.
reviewed by: Eugene Seneta
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 540057
zbMATH Keywords: explicit forms
zbMATH Keywords: stochastic matrices
zbMATH Keywords: ergodicity coefficient
zbMATH Keywords: eigenvalue bounds
zbMATH Keywords: Deutsch-Zenger numerical radius
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)90113-3
OpenAlex ID: W2077837299
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description: scientific article; zbMATH DE number 540057