### Stochastic Matrices and the Perron-Frobenius Theorem

René ThiemannArchive of Formal Proofs 2017.

### 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.

### BibTeX

@article{Stochastic_Matrices-AFP, author = {Ren\'e Thiemann}, title = {Stochastic Matrices and the Perron-Frobenius Theorem}, journal = {Archive of Formal Proofs}, month = nov, year = 2017, note = {\url{http://isa-afp.org/entries/Stochastic_Matrices.html}, Formal proof development}, ISSN = {2150-914x}, }