What is the general decomposition method to find stationary distribution of a transition matrix in Markov Chain?

Question

Based on the definition of transition matrix in Wikipedia, I assume that it is not symmetric but is it always diagonalizable? The solution is written using eigen decomposition. How if it is not diagonalizable? Will it be solved using singular value decomposition?

Answer 1

No, the transition matrix of a Markov chain is a stochastic matrix that is not diagonalizable in general. The Perron-Frobenius theorem asserts that if the matrix is irreducible, there is a unique probability invariant vector associated to the largest eigenvalue.

So for example you compute the characteristic polynomial, take its largest eigenvalue $\lambda$ and solve the equation $Px = \lambda x$ to find a vector with non-negative coordinates. You then divide by the sum of its coordinates to get a probability vector.