# eigenvalues of stochastic matrix

Theorem:
The spectrum of a stochastic matrix^{} is contained in the unit disc in the complex plane^{}.

###### Proof.

Let $A$ be a stochastic matrix and let $m$ be an eigenvalue^{} of $A$, with $v$ eigenvector^{}; then, for any self-consistent matrix norm $\parallel .\parallel $, we have:

$$\left|m\right|\parallel v\parallel =\parallel mv\parallel =\parallel Av\parallel \le \parallel A\parallel \parallel v\parallel ,$$ |

that is, since $v$ is nonzero,

$$\left|m\right|\le \parallel A\parallel .$$ |

Now, for a (doubly) stochastic matrix,

$${\parallel A\parallel}_{1}=\underset{j}{\mathrm{max}}\left(\sum _{i}\left|{a}_{ij}\right|\right)=1$$ |

whence the conclusion^{}.
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Title | eigenvalues of stochastic matrix |
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Canonical name | EigenvaluesOfStochasticMatrix |

Date of creation | 2013-03-22 16:18:02 |

Last modified on | 2013-03-22 16:18:02 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 7 |

Author | Andrea Ambrosio (7332) |

Entry type | Theorem |

Classification | msc 60G99 |

Classification | msc 15A51 |