How do i determine the number of stationary distributions that a Markov chain has if it is not irreducible or regular.
The transition matrix is
1 0 0 0.25 0.5 0.25 0 0 1
is there infinitely many?
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Yes, exactly. We have two absorbing states : the first and the third. If process starts in the first state, it stays here forewer. And the same for the third state. So both probability vectors $\pi_1=(1,0,0)$ (row vector) and $\pi_3=(0,0,1)$ provide stationary distributions : $\pi_1P=\pi_1$ and $\pi_3P=\pi_3$. And any linear combination of these vectors $\pi=p\pi_1+(1-p)\pi_3$ with $0\leq p\leq 1$ provides stationary distribution too: $$ \pi P= p\pi_1P+(1-p)\pi_3P =p\pi_1+(1-p)\pi_3 = \pi. $$