stationary distribution of a transition matrix

Question

How can I solve the stationary distribution of a finite Markov Chain? In other words, how can I estimate the eigenvectors of a transition matrix?

Answer 1

The Eigensystem[ ] command would be the way to go. Say you have a transition matrix:

trans = Transpose[{{1/6, 1/6, 4/6}, {0, 3/4, 1/4}, {1/10, 1/10, 8/10}}]

Then you would get the eigenvalues and eigenvectors as:

{eVals, eVecs} = Eigensystem[trans]

You can interpret these using

eVals // MatrixForm

and

eVecs // MatrixForm

In this case, for the transition matrix above, the eigenvector corresponding to the eigenvalue $1$ is the first row of the eVecs matrix, which is $\{ 0.12, 0.48, 1.\}$. You can check that this is true by evaluating

trans.{0.12, 0.48, 1.}

which indeed returns $\{ 0.12, 0.48, 1.\}$. To get the actual steady state distribution, you would need to normalize this, i.e., divide the vector by the sum of the elements

{0.12, 0.48, 1.}/Total[{0.12, 0.48, 1.}]