I have the below question: Background: There are 3 cities, K, W and G. When in W, the probability of the tourist going to K=p, G=0 and W= 1-p. When in K, probabilityfor G=p, W=1-p , and k=0. When in G, k=1-p,W= 0 and G=p. Question to answer: Assuming the start of the trip in W, what are the probabilities of being in W, K or G after n days? Using a *closed form formula.
My Attempt: I know that we get a matrix below which can be reduced using eigen values. However, the below matrix is not symmetrical so can there be a closed form formula? Closed form formulas are generally only for Toeplitz or symmetrical matrices (https://en.wikipedia.org/wiki/Toeplitz_matrices) We get a non-symmetrical matrix of $$ \begin{bmatrix} 1-p & 1-p & 0 \\ p & 0 & 1-p \\ 0 & p & p \\ \end{bmatrix} $$
My attempt: I found these to be the 3 eigen values of 1 and +√((ρ-ρ^2) )and -√((ρ-ρ^2) ) Is the answer just the decomposed matrix (https://en.wikipedia.org/wiki/Tridiagonal_matrix) Which is made up of the eigen vectors formed by the eigen values which are $$ \begin{bmatrix} 1 & 1 & 1 \\ p/(1-p) & \frac{p-1-\sqrt{p-p^2}}{(1-p)} & \frac{\sqrt{ρ-ρ^2}+p-1}{1-p} \\ \frac{P^2}{(1-p)^2} & \frac{\sqrt{p-p^2}+1-p}{(1-p)(p+\sqrt{p-p^2})} & \frac{\sqrt{p-p^2}-1+p}{(1-p)(p-\sqrt{p-p^2})} \\ \end{bmatrix} $$
If you need me to update, tell me.
EDIT: So I noticed that there is actually no solution for the Jordan decomposition. I think this is because the roots are complex $$ \begin{bmatrix} 1 & \sqrt{p-p^2} & -\sqrt{p-p^2} \\ \end{bmatrix} $$ Is this a correct conclusion.