probability, random walk, Markov chain question

Question

Let $P$ be a transition matrix for a regular Markov chain and let $w$ be it’s equilibrium vector. Show that $w$ has no zero entries.

Answer 1

Regular Markov chain means that all the entries of the transition matrix $P$ are positive. Let Markov chain have $n$ states. If $\pi _i$ is the equilibrium probability for state $i$, you can say:

$$ \pi _i = \sum _{j=1}^n\pi _j P_{ji} $$ Assume by contradition that $\pi _i=0$. Since all the entries of $P$ are positive, from equation above you can conclude that $\pi_j = 0$ for all $j = 1, \dots, n$ that is a contradition since we know that $\sum_{j=1}^n \pi_j = 1$.