Explicit forms for ergodicity coefficients and spectrum localization

Abstract

Explicit forms for orgodicity coefficients which bound the non-unit eigenvalues of finite stochastic matrices are known in the cases p = 1, ∞ when the l p norm is used [3,6,7]. The purpose of this report is to generalize these results to arbitrary real matrices (in particular square matrices with a real-valued eigenvalue), by giving a generalized (and to some extent simplified) synthesis of methods used hitherto. Alternative procedures are mentioned in the light of the framework established. The results are of particular relevance for a primitive nonnegative matrix, for which a bound results for the non-Perron-Frobenius eigenvalues.