### Abstract

Stochastic matrices are a convenient way to model discrete-time and
finite state Markov chains. The Perron–Frobenius theorem
tells us something about the existence and uniqueness of non-negative
eigenvectors of a stochastic matrix. In this entry, we formalize
stochastic matrices, link the formalization to the existing AFP-entry
on Markov chains, and apply the Perron–Frobenius theorem to
prove that stationary distributions always exist, and they are unique
if the stochastic matrix is irreducible.