# Markov Matrix with complex eigenvalues

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?

• What do complex eigenvalues of any real matrix indicate? See this question for one explanation. – amd May 19 at 20:23
• If in a Markov Matrix the sum of the columns is $1$, then the dominant eigenvalue is $1$ and the other eigenvalues (complex or real) are irrelevant. An eigenvector of eigenvalue $1$ is responsible for the Steady State – imranfat May 19 at 20:26
• If I start in the state $(1,0,0)^T$, I will not reach a steady state, right? – user560483 May 19 at 20:32
• I assume the Markov Matrix is a regular matrix, meaning one can leave from any state and arrive in any state, and ignoring "funny" examples like having only $1$'s and $0$'s in the matrix. Then the initial state doesn't matter, one will obtain the same Steady State.Moreover, in your provided example, did you calculate the eigenvalues? – imranfat May 19 at 20:34