The Perron–Frobenius theorem and the Hilbert metric

We start by stating a version of the Perron–Frobenius theorem. Let A be a d × d stochastic matrix, where here we use this to mean that the entries of A are non-negative, and every column sums to 1: Aij ∈ [0, 1] for all i, j, and ∑d i=1Aij = 1 for all j. Thus the columns of A are probability vectors. Such a matrix A describes a weighted random walk on d sites: if the walker is presently at site j, then Aij gives the probability that he will move to site i at the next step. Thus if we interpret a probability vector v as giving the probability of the walker being at site j with probability vj, then v 7→ Av gives the evolution of this probability under one step of the random walk. Now one version of the Perron–Frobenius theorem is as follows: If A is a stochastic matrix with A > 0 (that is, Aij > 0 for all i, j), then there is exactly one probability vector π that is an eigenvector for A. Moreover, the eigenvalue associated to this eigenvector is 1, the eigenvalue 1 is simple, and all other eigenvalues have modulus < 1. In particular, given any v ∈ [0,∞) we have Av → π exponentially quickly. The eigenvector π is the stationary distribution for the random walk (Markov chain) given by A, and the convergence result states that any initial