Burnside graphs, algebras generated by sets of matrices, and the Kippenhahn Conjecture

TL;DR

This paper introduces the use of Burnside graphs to analyze matrix-generated algebras, providing conditions for when these algebras are full, and constructs new counterexamples to the Kippenhahn conjecture.

Contribution

It develops a novel approach using Burnside graphs to study matrix algebras and presents new counterexamples to the Kippenhahn conjecture of size 8x8 and larger.

Findings

Established necessary and sufficient conditions for matrix sets to generate full algebras.

Developed a method to construct counterexamples to the Kippenhahn conjecture.

Produced new counterexamples of order 8x8 and above.

Abstract

Given a set of matrices, it is often of interest to determine the algebra they generate. Here we exploit the concept of the Burnside graph of a set of matrices, and show how it may be used to deduce properties of the algebra they generate. We prove two conditions regarding a set of matrices generating the full algebra; the first necessary, the second sufficient. An application of these results is given in the form of a new family of counterexamples to the Kippenhahn conjecture, of order $8 \times 8$ and greater.