Strong shift equivalence and positive doubly stochastic matrices

We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over R+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+R+ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over RR with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over RR are identical. We exhibit a class of 2×22×2 matrices, pairwise strong shift equivalent over R+R+ through 2×22×2 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×nn×n primitive matrix of positive trace that the set of positive n×nn×n matrices similar to it contains only finitely many SSE–R+R+ classes.