On explicit forms for ergodicity coefficients
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.
scientific article; zbMATH DE number 540057
scientific article