(Alternate proof of Theorem 11.8) Let P be the transition matrix of an ergodic Markov chain. Let...

Question:

(Alternate proof of Theorem 11.8) Let P be the transition matrix of an ergodic Markov chain. Let x be any column vector such that Px = x. Let M be the maximum value of the components of x. Assume that xi = M. Show that if pij > 0 then xj = M. Use this to prove that x must be a constant vector.