How can I get the symbolic steady state vector of a Markov Chain?
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Hello, does anyone know how to obtain the symbolic steady state vector (i.e. the long-term probability of each state) of this Markov Chain example in MATLAB?
At the end of this demonstration, it does not show how can I further get a steady state vector?
It will be very appreciated if you can help me with this problem.
John D'Errico on 7 Aug 2022
Edited: John D'Errico on 7 Aug 2022
Easy, peasy. For example, given a simple Markov process, described by the 3x3 transition matrix T.
T = [.5 .2 .3;.1 .4 .5;.1 .1 .8]
There are no absorbing states. We can see this is indeed the transition matrix of a Markov chain. One good test is the rows all sum to 1, and none of the elements are greater than 1, or less than zero.
What are the steady-state probabilities?
[V,D] = eig(T')
Take eigenvector that corresponds to the unit eigenvalue. In this case, it is the first eigenvector.
P = V(:,1)';
Normalize so the elements sum to 1.
format long g
P = P/sum(P)
Those are the steady state probabilites for this system. We can see that this does not change P.
I won't do your homework for you, but you can easily enough see how to proceed from here.