Powers of nonsymmetric stochastic matrices: comment on a paper by little, Ng, and Shaw

Abstract

In a recent paper, Little et al. point out that certain terms in the eigenvector expansion of a nonsymmetric stochastic matrix are unbounded. This appears to throw doubt on the usual assumption that only terms associated with the largest eigenvalues contribute significantly to high powers of the matrix. In this note we show that although individual terms can be unbounded, their sum is always well behaved, and since this is the relevant quantity, the eigenvalue criterion is still valid. An explicit bound on the sum is obtained for a 3 × 3 case, and for the general N × N case it is shown that the factors causing the unboundedness cancel.