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I found plenty of proofs online that a Markov matrix will have all its eigen values of modulus <=1. In the book on Introduction to Matrix analysis by Bellman, he also shows in section 8 of chapter 14 that for positive Markov matrices, the only eigen value of magnitude 1 can be 1 itself (and not -1). I beleive positive Markov matrices imply all their elements are positive and so exclude matrices with absorbing states.

But, what I've observed is that for all such matrices I've come across, there is just one eigen value, 1 and all other eigen values are <1 in magnitude.

In fact, I've observed this for Markov matrices with just one absorbing state as well.

Is there a way to prove or refute (with counterexamples or otherwise) the above two claims?

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To make the question answered I note that a Markov matrix $$\begin{pmatrix} 1 & 0 & 0\\ 0 &0 & 1\\ 0 &1 & 0\end{pmatrix}$$ corresponding to a chain with just one absorbing state should refute the claim, because it has eigenvalues $1$ and $-1$

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