0
$\begingroup$

I'm a computer science student, and I'm really stuck in this lemma since I cant find out anything about the relation between a absorbent state (in a Markov chain $M$, a absorbent state is when the value $M_{ii} = 1$ and the rest of the values $M_{ij} = 0$ ) and its eigenvalue.

Lenma: For each absorbent state of a Markov chain, there are a eigenvector associate to the eigenvalue $\lambda = 1$.

I need this lemma to demonstrate the follow theorem:

Theorem In a Absorbent Makov Chain, the eigenvalue $\lambda = 1$ has a geometric multiplicity equal to the number of absorbent states (The numbers of $M_{ii} = 1$) and all the otres eigenvalues are $|\lambda| < 1$.

I've searching for about 6 months and I cant find anything about this or the theorems above. Any help is welcome since is the only one thing to end my bachelor thesis.

$\endgroup$
0
$\begingroup$

HINT: Think about the relationship between the “immediate future” state probability vector of your system and the present state probability vector using Markov chain theory.

If you hit an absorbent state, how would that relationship simplify? Could that relationship resemble something that is related to the definition of eigenvalues?

$\endgroup$

This site is temporarily in read only mode and not accepting new answers.

Not the answer you're looking for? Browse other questions tagged .