Let be a stochastic matrix (i.e. and for all ). Show that is an eigenvalue of this matrix and all its eigenvalues satisfy .

Since, is a stochastic matrix we can apply the Ergodic theorem which tells us that there exists with such that as . Thus, we see that .

To show that all eigenvalues of have a magnitude less than , note that

As a result, if an eigenvalue for an eigenvector , then . Similarly, if , then . Therefore, all eigenvalues must satisfy .