the Perron-Frobenius theorem

Why does a transition matrix always have an equilibrium, and why is it unique and reachable? The deep answer comes from linear algebra, not probability: the Perron-Frobenius theorem, a powerful result about matrices with non-negative entries. It is the structural reason behind the existence and uniqueness of the stationary distribution and the convergence of P^n.

The theorem says that an irreducible matrix with non-negative entries has a largest eigenvalue that is real, positive, and simple (no ties), with an eigenvector whose entries are all strictly positive — and no other eigenvector can be chosen all-positive. For a stochastic matrix P (rows summing to 1), that leading eigenvalue is exactly 1, all other eigenvalues have magnitude at most 1, and the positive left eigenvector, normalised to sum to 1, is precisely the stationary distribution pi solving pi P = pi. So 'a unique equilibrium exists with positive weight on every state' is just Perron-Frobenius applied to an irreducible chain.

It also explains convergence and its speed. If the chain is aperiodic, eigenvalue 1 is the strict champion: every other eigenvalue has magnitude strictly below 1, so their contributions to P^n decay, leaving only the stationary part — hence P^n converges. The second-largest eigenvalue magnitude controls how fast: the gap between 1 and it sets the mixing rate. A periodic chain has other eigenvalues also on the unit circle (like -1), which is exactly why it oscillates and fails to converge. PageRank is Perron-Frobenius made famous: Google's ranking is the positive Perron eigenvector of a giant stochastic matrix of the web.