Describing the Numerical Range and $C$-Numerical Range of Matrices via Their Unitary Orbits and the Joukowsky Transform

TL;DR

This work extends the Elliptical Range Theorem beyond 2x2 matrices by employing a Joukowsky-transform framework to describe the numerical range of a 3x3-matrix class and provides an alternate Specht-based proof leveraging unitary equivalence. It then delivers explicit C-numerical range descriptions, including an ellipse for 2x2 cases and a rank-1 generalisation via a multidimensional union-of-ranges construction. The methodology emphasizes unitary orbits, support functions, and Minkowski sums to obtain geometric characterisations that avoid heavy Kippenhahn polynomial calculations. Collectively, the results unify several strands of numerical-range theory with practical, constructive descriptions potentially relevant to questions like Crouzeix's conjecture and higher-rank generalisations.

Abstract

We generalise the Elliptical Range Theorem to characterise the numerical range of matrices belonging to a subspace of the space of \(3 \times 3\) matrices. Using Specht's Theorem, which characterizes when two matrices are unitarily equivalent, we then provide a novel proof of the Elliptical Range Theorem. Finally, we give an explicit description of the $C$-numerical range for $2 \times 2$ matrices and for rank-one matrices.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 2.1
• proof
• Corollary 2.2
• Example 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Example 2.7