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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.

 

   AFP entry

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},
}
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