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?

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    $\begingroup$ What do complex eigenvalues of any real matrix indicate? See this question for one explanation. $\endgroup$ – amd May 19 at 20:23
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    $\begingroup$ 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 $\endgroup$ – imranfat May 19 at 20:26
  • $\begingroup$ If I start in the state $(1,0,0)^T$, I will not reach a steady state, right? $\endgroup$ – user560483 May 19 at 20:32
  • $\begingroup$ 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? $\endgroup$ – imranfat May 19 at 20:34

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