Spectrally arbitrary pattern extensions

TL;DR

This paper introduces a graphical triangle extension technique to construct and analyze spectrally arbitrary matrix patterns, expanding the known families and providing new methods for pattern extension and inertial arbitrariness.

Contribution

It presents a novel triangle extension method for building spectrally arbitrary patterns from lower order patterns and extends these ideas to inertially arbitrary patterns.

Findings

Triangle extension constructs new spectrally arbitrary patterns.

The method reveals relationships among known patterns.

Extensions apply to inertially arbitrary patterns, enlarging their classes.

Abstract

A matrix pattern is often either a sign pattern with entries in {0,+,-} or, more simply, a nonzero pattern with entries in {0,*}. A matrix pattern A is spectrally arbitrary if for any choice of a real matrix spectrum, there is a real matrix having the pattern A and the chosen spectrum. We describe a graphical technique, a triangle extension, for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns. These methods provide a new way of viewing some known spectrally arbitrary patterns, as well as providing many new families of spectrally arbitrary patterns. We also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns. We then provide an additional extension method for zero-nonzero patterns.