What kind of questions can I answer if I have a transition matrix of a system I understand how a transition matrix looks. It is a matrix of probabilities of moving between one state to another.  After I am given such a matrix, what are some questions that I can answer about the system?
Basically, what do I do with this transition matrix? Can it be used for some specific purpose or is it an end product in itself?
 A: A random walk defined by the transition probabilities in this matrix, $A$, will be completely described by starting position (or perhaps some starting distribution) and this matrix.  Therefore using this matrix you can answer lots of great questions, for example:


*

*If start in position $s_0$, or if my initial position is a random variable represented by the distribution $P(S)$, what is the distribution of my position after $n$ moves?  This is obtained by taking your starting distribution, representing it as a vector of probabilities, $v_0$, and computing the product:
$$
A^n \cdot v_0.
$$
In the case where you know exactly where you begin $v_0$ should be a standard basis vector with a 1 in the corresponding position.

*What is the stationary distribution of this process?  Under certain conditions, Markov Processes have stationary distributions.  Essentially, if my current position is a random variable with a certain distribution, my next position will follow the same distribution.  This distribution of course corresponds to an eigenvector with eigenvalue 1.
$$
A \cdot v = v.
$$
