# Number of stationary distributions of a Markov chain

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?

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: $$\pi P= p\pi_1P+(1-p)\pi_3P =p\pi_1+(1-p)\pi_3 = \pi.$$