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,

$${\parallel A\parallel }_1=\underset{j}{\max }\left(\sum _i\left|a_{ij}\right|\right)=1$$

whence the conclusion.∎