# How to bound the second largest eigenvalue of a transition matrix of a non-irreducible Markov chain?

I have found several bounds (e.g., Cheeger, Poincare) for the case that the Markov chain is irreducible and reversible, however my Markov chain has one absorbing state. Any bound would be helpful, but I am especially interested in an upper bound (less than 1 of course).

Thanks!

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jstor.org/stable/2959625 –  alex Feb 7 '11 at 2:02