1. The problem statement, all variables and given/known data Given an initial distribution state vector that represents the probability of the system to be in one of its states. Also given a Markov transition matrix. How to calculate the state vector of the system after n-transition? 2. Relevant equations Assuming the initial state vector is a column vector x0, in literature it is considered as a row vector x0T. The state vector after the first transition will be: $$P x_0=x_1$$ where P is the Markov transitional matrix. After n-transitions, $$P^n x_0=x_n$$ 3. The attempt at a solution I tried to decompose P for an easier calculation using singular value decomposition, assuming that it is asymmetric matrix. $$P=UΣV^T$$. $$P^n=(UΣV^T)^n=(U)^n (Σ)^n (V^T)^n$$. I know that Σn is a diagonal matrix. Also I know that Un=I. But, I do not know the behavior of (VT)n.