Finite Markov chain theory and its connection with matrix theory

To study finite Markov chains, we begin with the theory of order relations to classify states and chains. Then we define various functions on the chain and use the theory of probability and statistics to find their means and variances. Throughout the whole study, however, the connection viith matrix theory is built-in since a finite Markov chain can be represented as a stochastic matrix. Many questions concerning finite Markov chains can be answered, directly or indirectly, by investigating only two kinds of chains: absorbing Markov chains and regular Markov chains. Though these chains are different, the studies of these chains offer many striking similarities.