is m(x) for each , j in S = 2, N}.

Theorem 5.1 Suppose P is the 1-step transition probability matrix of a Markov chain on a finite state space. Also suppose P is diagonalizable over the real numbers. Then

Remarks.

take values in the interval [-1,1]. Moreover, one of these roots is always 1. If the steady state distribution exists, the roots are in the interval (-1,1] and the coefficient of root 1 is the steady state probability of the Markov chain.