[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]

Let [math]\mat{P}[/math] be the transition matrix of an ergodic Markov chain. Let [math]\mat{w}[/math] be a fixed probability vector (i.e., [math]\mat{w}[/math] is a row vector with [math]\mat {w}\mat {P} = \mat {w}[/math]). Show that if [math]w_i = 0[/math] and [math]p_{ji} \gt 0[/math] then [math]w_j = 0[/math]. Use this to show that the fixed probability vector for an ergodic chain cannot have any 0 entries.