question about ergodic matrices

An ergodic matrix is a finite square matrix, M, with non-negative entries = such that there exists a k so that I + M + M^2 + ,,, + M^k >> 0, meaning t= hat all entries are strictly positive. One example is | 0 0 1 1 | | 0 0 1 0 | | 1 1 0 0 | | 1 0 0 0 1

If the columns are probability vectors then these matrices are Markov chain= s. This allows the process to be periodic as the example above has period =

1. Even powers have Fibonacci numbers in the upper right and lower left 2 b= y 2 blocks while odd powers have Fibonacci numbers in the upper left and lo= wer right 2 by 2 blocks so in this case k =3D 1 and m + m^2 >> 0. If there = is a k so that m^k >> 0 then the matrix is strictly ergodic. A strictly erg= odic state can, given a finite amount of time, get from any state to any ot= her. An ergodic process visits every state, but the set of times that (begi= nning at i) it visits j with positive probability is a subset of a lattice.