# 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 $\left\|.\right\|$, we have:

 $\left|m\right|\left\|v\right\|=\left\|mv\right\|=\left\|Av\right\|\leq\left\|A% \right\|\left\|v\right\|,$

that is, since $v$ is nonzero,

 $\left|m\right|\leq\left\|A\right\|.$

Now, for a (doubly) stochastic matrix,

 $\left\|A\right\|_{1}=\max_{j}\left(\sum_{i}\left|a_{ij}\right|\right)=1$

whence the conclusion. ∎

Title eigenvalues of stochastic matrix EigenvaluesOfStochasticMatrix 2013-03-22 16:18:02 2013-03-22 16:18:02 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 7 Andrea Ambrosio (7332) Theorem msc 60G99 msc 15A51