Estimates for eigenvalues of stochastic matrices

Abstract

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r2, …, rN} be the eigenvalues of stochastic matrix X of size N × N. We will present in this paper a simple necessary and sufficient condition for X such that |rj | < 1, j = 2, …, N. Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.