Inequalities for functions of transition matrices

The paper consists of two parts. In the first part, we consider two matrices that appear in the literature in the study of irreducible Markov chains. The first matrix N is equal to the mean first passage of the Markov chain except on the diagonal where N vanishes. The other matrix K is equal to , where J is the all-1 matrix, A is the identity minus the transition matrix of the Markov chain, and is the diagonal matrix whose diagonal entries are the corresponding diagonal entries of the group inverse of A. Both N and K are known to be invertible. We show that the diagonal entries of N-1 and of K-1 are strictly negative in sufficiently high dimensions (⩾3 for N and ⩾4 for K). These results lead to a number of inequalities of independent interest, one of which we study in greater detail probabilistically. In the second part of the paper, we address a problem raised by Kemeny and Snell of determining whether a given Markov chain is primitive only from its first mean passage matrix, without having to compute the transition matrix. We derive several simple conditions of the mean first passage matrix which are helpful in determining whether the corresponding transition matrix is primitive.