# Markov chain: Relation between absorbent states and its eigenvalue.

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.