#### Stochastic Matrices and the Perron-Frobenius Theorem

René Thiemann

Archive 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.

``@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},}``
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