If I have the $Q$ matrix, can I somehow find the $P$ matrix?

Question

I have this $Q$ matrix $$\left[\begin{array}{rrrr} -6& 1& 2& 3\\ 4& -15& 5& 6\\ 7& 8& -24& 9\\ 10& 11& 12& -33 \end{array}\right]$$

I was just wonering if there is any way to change the $Q$ matrix to transition $P$ matrix?

Answer 1

My guess is that by $Q$ matrix you mean the generator of a continuous-time Markov chain where the transition matrix for time $t$ is the matrix exponential $T(t) = \exp(t Q)$ and you want $P = T(1) = \exp(Q)$, the transition matrix for a discrete-time Markov chain such that $P^k = T(k)$ for positive integers $k$. There are lots of ways to compute it. See in particular Moler and val Loan, "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later".