How to compute stationary distribution for martrix with more than 1 closed class

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SUMMARY

The discussion focuses on computing the stationary distribution for a transition matrix with more than one closed class. The provided matrix has an absorbing state, specifically the third state, which necessitates rewriting the matrix in canonical form. To find the stationary distribution, users should solve the eigenvector equation xA = x, resulting in two linearly independent eigenvectors due to the presence of two closed classes: {1,4} and the absorbing state. Valid linear combinations can be determined by ensuring positivity and that the components sum to one.

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sam48
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Hi,

Thank you in advance if anyone can answer this question.

How any stationary distributions exists in below matrix and what are the value

[
.5 0 0 .5
.25 .5 .25 0
0 0 1 0
1/6 0 0 5/6
]

Any information regarding how to compute stationary distribution in a martrix with more than 1 closed class would be appreciated.
regards,
 
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Your chain has an absorbing state (the third state, since the 3,3 entry is 1). Write this matrix in canonical form and analyze it with the standard methods.
 


One way is just to solve the eigenvector equation xA=x - since your example has 2 closed classes (the other being {1,4}), there will be two LI eigenvectors. Positivity and summation to 1 will tell you which linear combinations are valid.
 

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