How to solve forward equation for a continuous-time Markov chain? 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!

For a generic matrix there is no simpler expression for the exponential. But there are many ways to calculate the exponential of a matrix. I suggest that you ask a new question with an explicit example (possibly with parameters) to get a feel for the method.
