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# Markov chain help

1. Just a quick couple of questions:
1
What is meant by the 'equilibrium distribution vector' concerning a Markov chain of probabilities?

Take this example:

1---------2----------3

"An object moves along the graph shown above, so that at each step it moves with equal probability to a neighbouring point."

a) Write down the transition matrix P for this Markov process (i think i've got this right, so i'm not bothered about this one)
b) Calculate the equilibrium distribution vector for this Markov process.

2
How do you calculate the limit as n -> infinity of A^n for a square matrix A?
2. From my rather hazy memory, an equilibrium (or invariant) distribution vector is a stochastic row vector with .
3. (Original post by generalebriety)
From my rather hazy memory, an equilibrium (or invariant) distribution vector is a stochastic row vector with .
Thanks, that makes sense now.
4. In general, finding is tricky (even assuming the limit exists).

But barring a few pathological cases(*), if M is a stochastic matrix, and is an invariant vector for M, then . Which means M^n converges to the matrix with all columns equal to .

(*) I think it always works as longs as M has no elements of size 1, but that's intuition and hope speaking rather than detailed knowledge.

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