I want to compute the infinitesimal generator of the Markov chain associated with the following graph:
The transition matrix if the following: $$ M = \left( \begin{matrix} \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{matrix} \right) $$
Is it correct that the infinitesimal generator is the following matrix:
$$ A = \left( \begin{matrix} -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & -1 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & -1 & 0 \\ 0 & 0 & \frac{1}{2} & -\frac{1}{2} \end{matrix} \right) $$
From $A = \lambda(M − I )$?
In other words, is $\lambda=1$? If so, why?