Inequalities for functions of transition matrices

• Published in: Linear algebra and its applications 2012-01, Vol.436 (2), p.335-348
• Main Authors: Ben-Ari, Iddo, Neumann, Michael, Pryporova, Olga
• Format: Article
• Language: English
• Subjects: Absorbing states; Algebra; Exact sciences and technology; Inverse M-matrices; Linear and multilinear algebra, matrix theory; Markov chains; Markov processes; Mathematics; Mean first passage times; Nonnegative matrices; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stationary distribution
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2011.04.044

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 JA d # - A , where J is the all-1 matrix, A is the identity minus the transition matrix of the Markov chain, and A d # 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.