Refined Inertia of Matrix Patterns

TL;DR

This paper investigates how the placement of zeros in matrices influences their eigenvalues, introducing new classes of patterns with specific inertial properties and characterizing small-order cases.

Contribution

It introduces refined inertially arbitrary patterns, distinguishes them from spectrally arbitrary patterns, and characterizes small-order cases with minimal nonzero entries.

Findings

Existence of inertially arbitrary patterns with no 2-cycle digraphs

Development of a class of refined inertially arbitrary but not spectrally arbitrary patterns

Complete characterization of order three and minimal order four patterns

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.