Classification of joint numerical ranges of three hermitian matrices of size three

TL;DR

This paper classifies the possible shapes of joint numerical ranges of three 3x3 Hermitian matrices, revealing only ten configurations of faces and illustrating these with examples.

Contribution

It provides a complete classification of the face structures of joint numerical ranges for three 3x3 Hermitian matrices, identifying ten possible configurations.

Findings

Joint numerical range is generically a 3D oval.

Faces are segments or filled ellipses.

Only ten face configurations are possible.

Abstract

The joint numerical range $W(F)$ of three hermitian $3$-by-$3$ matrices $F=(F_1,F_2,F_3)$ is a convex and compact subset in $\mathbb{R}^3$. Generically we find that $W(F)$ is a three-dimensional oval. Assuming $\dim(W(F))=3$, every one- or two-dimensional face of $W(F)$ is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple $F$ for each class and illustrate $W(F)$ using random matrices and dual varieties.