### 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.

### License

BSD License

### Topics

- Mathematics/Algebra
- Computer science/Automata and formal languages

### Session Stochastic_Matrices

- Stochastic_Matrix
- Stochastic_Vector_PMF
- Stochastic_Matrix_Markov_Models
- Eigenspace
- Stochastic_Matrix_Perron_Frobenius

### Depends on

- Matrices, Jordan Normal Forms, and Spectral Radius Theory
- Markov Models
- Perron-Frobenius Theorem for Spectral Radius Analysis

## Cite

×

Stochastic_Matrices-AFP

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

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