Refined Inertia of Matrix Patterns
Rings and Algebras
2016-11-28 v1 Combinatorics
Abstract
We explore how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. We demonstrate that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. We develop a class of patterns that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. We include a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.
Cite
@article{arxiv.1611.08217,
title = {Refined Inertia of Matrix Patterns},
author = {Jonathan Earl and Kevin N. Vander Meulen and Adam Van Tuyl},
journal= {arXiv preprint arXiv:1611.08217},
year = {2016}
}