I have noted that the steady state probabilities of an irreducible Markov chain can be written as fractions that have the same denominator. Is there any result about this property? What does this denominator represent?
Example
Let us consider a Markov chain with the following transition probabilities:
$P = \left[\begin{array}{ccc} p & 1-p & 0 \\ q & 0 & 1-q \\ 1 - r & 0 & r \\ \end{array}\right]$
The steady state distribution is:
$\left(\dfrac{1 - r}{pq + pr - 2p - q - 2r + 3}, \dfrac{1 - p - r(1 - p)}{pq + pr - 2p - q - 2r + 3}, \dfrac{1 - p - q(1 - p)}{pq + pr - 2p - q - 2r + 3}\right)$
As we can see, all these fractions have $pq + pr - 2p - q - 2r + 3$ in the denominator. I've found that this observation is "always" preserved.
Thanks in advance for your help.
The steady state is the solution of a system of linear equations. As such, it can be obtained by dividing suitable determinants, which themselves are polynomial expressions in the given entries. As it is always the same deteminant that occurs in the denominator, your observation follows.