What properties does a Markov matrix (with real entries) with complex eigenvalues have?
For example, consider this matrix: $$\begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ \end{pmatrix}.$$
If I start in the state $(1,0,0)^T$, this does not have a steady state, right?