# 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!

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Are you familiar with eigendecomposition? – Emre May 10 '11 at 0:14
@Emre: Thanks! Is eigendecomposition one usual way to solve it? What if the process is given as a linear pure birth problem, i.e. Yule process, with rates $k\lambda, k=1,2,..n$ where values of $n$ and $\lambda$ are unknown? Can the rate matrix with unknown constants still be eigendecomposed? – Tim May 10 '11 at 0:26

## 1 Answer

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.

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