From my Markov Chain, I have a generator matrix $G$=
\begin{bmatrix} -20 & 20 & 0 \\ 12 & -32 & 20 \\ 0 & 12 & -12 \end{bmatrix}
and I wish to find its stationary distribution $\pi=(\pi_0, \pi_1, \pi_2)$, which is solved by
$\pi G = 0$
and I also know that $\pi_0+\pi_1+\pi_2=1$.
How can I compute this using the generator matrix? Wouldn't I just end up equating to zero? I've seen a few examples on the website but they did this using the probability transition matrix $P_t$ instead; $\pi P_t = \pi$.
For simplicity, let's write the $\pi's$ as $(A,B,C)$.
Wouldn't I just have the following set of equations? \begin{align} \label{eqn:eqlabel} \begin{split} -20A+12B=0 \\ 20A-32B+12C=0 \\ 20B-12C=0 \end{split} \end{align}
That’s correct, but include the normalizing equation $A+B+C=1$ in the system.
In practical terms, you can simply compute a null vector of $G^T$ and then normalize it to make it stochastic.