Strong shift equivalence and positive doubly stochastic matrices

Abstract

We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over R+ to a positive doubly stochastic matrix through matrices of the same size. We also prove that every positive stochastic matrix is strong shift equivalent over R+ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over R with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over R are identical. We exhibit a class of 2×2 matrices, pairwise strong shift equivalent over R+ through 2×2 matrices, for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size. In contrast, we show for any n×n primitive matrix of positive trace that the set of positive n×n matrices similar to it contains only finitely many SSE–R+ classes.