Title: A matrix sequence {Gamma(A^m)}_{m=1}^infty might converge even if the matrix A is not primitive
Speaker: Boram Park, DIMACS, Rutgers University
Date: Tuesday, November 20, 2012 2:00pm
Location: Hill Center, Room 525, Rutgers University, Busch Campus, Piscataway, NJ
It is well-known that, for an irreducible Boolean (0,1)-matrix A, the matrix sequence {A^m}_{m=1}^infty converges if and only if A is primitive. In this paper, we introduce an operation Gamma on the set of Boolean (0,1)-matrices such that a matrix sequence {Gamma(A^m)}_{m=1}^infty might converge even if the matrix A is not primitive. Given a Boolean (0,1)-matrix A, we define a matrix Gamma(A) so that the (i,j)-entry of Gamma(A) equals 0 if for i neq j, the inner product of the ith row and jth row of A is 0 and equals 1 otherwise. The aim of this talk is to study the convergence of {Gamma(A^m)}_{m=1}^infty for a Boolean (0,1)-matrix A whose digraph has at most two strong components.. We show that {Gamma(A^m)}_{m=1}^infty converges to a very special type of matrix as m increases if A is an irreducible Boolean matrix. Furthermore, we completely characterize a Boolean (0,1)-matrix A whose digraph has exactly two strongly connected components and for which {Gamma(A^m)}_{m=1}^infty converges, and find the limit of {Gamma(A^m)}_{m=1}^infty in terms of its digraph when it converges. We derive these results in terms of the competition graph of the digraph of A.
See: http://math.rutgers.edu/seminars/allseminars.php?sem_name=Discrete%20Math