Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but not sufficient that P has real eigenvalues.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 14EQ
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Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible
in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly
positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but
not sufficient that P has real eigenvalues.
Transcribed Image Text:Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but not sufficient that P has real eigenvalues.
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