[SOLVED] Finding the stationary distribution of a markov chain using the eigenvectors of the transition matrix

Finding the stationary distribution of a markov chain using the eigenvectors of the transition matrix

I have been trying to find the stationary distribution pi for a transition matrix P My example 5x5 Matrix P results in an eigenvector I get by doing the following: Example P matrix:

0.5 0.2 0.3
0.6 0.2 0.2
0.1 0.8 0.1

eigenvalue, eigenvector = eigen(P)

I get a 5x1 Vector for the eigen value with the last element being eigenvalue of 1. I believe that pi should be some multiple of the eigenvector associated with that eigenvalue of 1. However, it seems like the eigenvectors I have are not a proportion of my pi. What am I doing wrong?

EDIT: I have found that I should be doing eigen(P'). This at least gives me the correct eigenvectors, however how do I find the right multiple of the eigenvector to find pi without knowing in advance what pi is?

Solution

So far the post is correct. The way to find the proportion, or scale, is to use the fact that the sum of the stationary distribution pi is always 1.

scale = 1/sum(eigenvector) # for the relevant eigenvector not the whole matrix
pi = eigenvector * scale