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Stationary distribution of mixture of Markov Chain with "complete" Markov Chain

I already asked this question in StackExchange, but found little attention. So I'm just going to copy-paste my original question here. Let $P$ be a stochastic matrix (of an irreducible Markov Chain) ...
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Simple random walk on a discrete torus - the eigensystem, reference

My problem concerns finding a reference in which the formulae for the eigenvalues and the corresponding eigenvectors ($n$ linearly independent eigenvectors!) for the transition matrix of a simple ...
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