On the structure of stochastic matrices with a subdominant eigenvalue near 1 Original Research Article

An n × n irreducible stochastic matrix P can possess a subdominant eigenvalue, say λ2(P), near λ = 1. In this article we clarify the relationship between the nearness of these eigenvalues and the near-uncoupling (some authors say “nearly completely decomposable”) of P. We prove that for fixed n, if λ2(P) is sufficiently close to λ = 1, then P is nearly uncoupled. We then provide examples which show that λ2(P) must, in general, be remarkably close to 1 before such uncoupling occurs.