Surfaces and hypersurfaces as the joint spectrum of matrices

TL;DR

This paper explores the Clifford spectrum of multiple Hermitian matrices, providing computational methods to generate and analyze examples where the spectrum forms complex geometric structures.

Contribution

It introduces computational techniques to explicitly calculate the Clifford spectrum for three or four matrices, expanding the set of concrete examples studied.

Findings

Clifford spectrum can form two-dimensional manifolds for small matrices

Computational methods enable explicit calculation of the Clifford spectrum

Examples of Clifford spectra for multiple matrices are systematically generated

Abstract

The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operators. While it has been know that for examples as small as three $2$-by-$2$ matrices the Clifford spectrum can be a two-dimensional manifold, few concrete examples have been investigated. Our main goal is to generate examples of the Clifford spectrum of three or four matrices where, with the assistance of a computer algebra package, we can calculate the Clifford spectrum.