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?

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    – Verbeia
    Oct 4, 2012 at 7:46

1 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


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.}]
  • $\begingroup$ nice answer! Another way to do the last step would be Normalize[{0.12, 0.48, 1.}, Total]. $\endgroup$ Oct 4, 2012 at 13:46
  • $\begingroup$ yes, it works, thanks. why do you need to transpose? $\endgroup$
    – whynot
    Oct 4, 2012 at 19:58
  • $\begingroup$ As commonly defined, a stochastic matrix has row sum equal to $1$, and has a left eigenvector corresponding to the unit eigenvalue. Mathematica computes right eigenvectors by default. So what the above does is to calculate the right eigenvector of $A'$, which is the desired left eigenvector of $A$. $\endgroup$
    – bill s
    Oct 5, 2012 at 5:59

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