calculating exact probability using eigenvalue and eigenvector

Question

I have markov chain model of state A and B where \begin{equation} \begin{pmatrix} A & B \end{pmatrix} \begin{pmatrix} 0.5 & 0.5 \\ 1/3 & 2/3 \end{pmatrix} \end{equation} is a probability matrix (i.e. A->A=0.5 and B->A=1/3). After many runs ($\lim_{t \rightarrow \infty}) $ the probability of being at status A is 0.4 and at B is 0.6. However I would like to develop a function $f(t)$ to calculate exact probability of being at state A in each run. (clue: It requires understanding of eigenvalue and convector)

Answer 1

If you have a Markov chain with intial state (as a row vector, so evolve on the right) $\pi$ and transition matrix $P$, then the state $k$ time steps later is given by $\pi P^k$.

You can compute $P^k$ in a variety of ways -- induction, eigendecomposition, etc. depending on the $P$.