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arXiv · 1707.06748

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

Ben Lawrence
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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.

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Ben Lawrence. 2017-07-21. Burnside graphs, algebras generated by sets of matrices, and the Kippenhahn Conjecture. https://arxiv.org/abs/1707.06748

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