Nonnegative matrices with doubly stochastic powers

Let A be a nonnegative irreducible square matrix, and let m be an integer greater than one. Then it is possible to obtain the following necessary and sufficient conditions for A[raised m] to be doubly stochastic while A is not doubly stochastic. First, A is cyclic of index h, where h and m are not relatively prime. Also, there exist positive numbers [beta][lowered i] and [zeta][lowered i], i=l,...,h, such that the matrices A[lowered i] in a Frobenius normal form of A are respectively [beta][lowered i]/[beta][lowered i+1] stochastic, indices modulo h, and such that the matrices A[raised T, lowered i] are [zeta][lowered h-i]/[zeta][h-i+1] stochastic respectively, indices modulo h...