[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 a regular transition matrix and let [math]\mat{w}[/math] be the unique non-zero fixed vector of [math]\mat{P}[/math]. Show that no entry of [math]\mat{w}[/math] is 0.