Ergodic Markov chains and eigenvalues

Question

I just read on wikipedia that a way to check whether a Markov chain is ergodic is to compute the eigenvalues of the transition matrix, and if those are all (except for 1) less than 1, then the chain is ergodic.

But consider the following matrix:

\begin{bmatrix}0&1&0&0\\0&0&0&1\\0&1&0&0\\0&0&0&1\end{bmatrix}

It is not ergodic, but the eigenvalues are $(1,0,0,0)$.

What am I missing?

Answer 1

I think the only thing you're missing is that the columns of a transition/stochastic matrix must sum to 1 (or the rows must sum to 1 for it to be right stochastic).