Research Initiation Award Grant: Investigating the combinatorial structure of special classes of matrices and graphs

Morehouse College, Atlanta GA

Investigators

Abstract

The Research Initiation Award entitled - Investigating the combinatorial structure of special classes of matrices and graphs - has the goal to find a test that will determine the eventual nonnegativity of reducible matrices and determine eventual properties of other classes of matrices. In dynamical systems, one is frequently interested in qualitative information regarding state evolution. Due to physical and modeling constraints arising in applications, it is of interest to impose or consider conditions for nonnegativity of the states. Such applications are directly linked to the problem of understanding the behavior of A^k as k increases. The objectives in this project will transform how we answer open questions about the nonnegativity and reducibility of large powers of matrices. An eventual property of a matrix M is a property that holds for all powers M^k, k >= k0, for some positive integer k0, the power index. Eventually positive matrices and eventually nonnegative matrices have applications to control theory and have been studied since their introduction in 1978. For a fixed n, the power index of an eventually positive or eventually nonnegative n x n matrix may be arbitrarily large, so it is not possible to show a matrix is not eventually positive or not eventually nonnegative by computing powers. Perron-Frobenius theory shows several ways to test for eventual positivity and in 2010 Hogben found a test for eventual nonnegativity for matrices that are not eventually reducible. The objective of this research is to investigate the remaining class of eventually nonnegative matrices that are not well understood.