Given the transition rate matrix of a CTMC as $G$, I was wondering how the forward equation P'(t) = P(t) G, P(0)=I is usually solved for the transition matrix $P(t)$?
Some book says the solution has the form $P(t) = exp\{tG\}$. Since exponential of a matrix is defined as a series form, I don't know if such form for solution can be simplified, and be helpful in determining the distribution given the beginning/ending state i.e. a row/column vector in $P(t)$.
Thanks and regards!