How to get transition matrix of markov process?

Question

I am monitoring a Markov process with ~21 states. I know all the states, initial state and what states transitions can/cannot be, so that zero elements of transition matrix are known.

I know the probability matrix P after k number of transitions and I want to calculate the transition matrix.

According to formula I need to perform k-root extraction from P, but I do not know how exactly.

Please guide me.

Answer 1

You could try Rice iteration to calculate the transition matrix $T=P^{1/k}$, which should work if $P$ is symmetric positive definite. The iteration starts with $T_0=0$ and then proceeds via

$$T_{n+1}=T_n + \frac{1}{k}(P-T_n^k)$$