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}
Best Answer
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.