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arXiv · 1603.06569

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

Konrad Szymański·Stephan Weis·Karol Życzkowski
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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.

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Konrad Szymański, Stephan Weis, Karol Życzkowski. 2016-03-20. Classification of joint numerical ranges of three hermitian matrices of size three. https://doi.org/10.1016/j.laa.2017.11.017

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