[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

(Alternate proof of Theorem) Let [math]\mat{P}[/math] be the transition matrix of an ergodic Markov chain. Let [math]\mat{x}[/math] be any column vector such that

[math]\mat{P} \mat{x} = \mat{ x}[/math]. Let [math]M[/math] be the maximum value of the components of [math]\mat{x}[/math]. Assume that [math]x_i = M[/math]. Show that if [math]p_{ij} \gt 0[/math] then [math]x_j = M[/math]. Use this to prove that [math]\mat{x}[/math] must be a constant vector.