Bordering for spectrally arbitrary sign patterns

TL;DR

This paper introduces a matrix bordering method to construct larger spectrally arbitrary sign patterns from smaller ones, extending previous techniques and enabling new constructions of inertially arbitrary patterns.

Contribution

It presents a generalized bordering technique for creating larger spectrally arbitrary sign patterns, expanding the toolkit for pattern construction.

Findings

Developed a recursive bordering method for sign pattern construction.

Extended the triangle extension method to broader cases.

Enabled construction of inertially arbitrary sign patterns.

Abstract

We develop a matrix bordering technique that can be applied to an irreducible spectrally arbitrary sign pattern to construct a higher order spectrally arbitrary sign pattern. This technique generalizes a recently developed triangle extension method. We describe recursive constructions of spectrally arbitrary patterns using our bordering technique, and show that a slight variation of this technique can be used to construct inertially arbitrary sign patterns.