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

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.