Matrices with invariant by rotation numerical ranges

TL;DR

This work characterizes d×d complex matrices A whose numerical range W(A) is invariant under a rotation by angle $2\pi/d$ about the origin. It introduces condition $(C_d)$, linking a real polynomial $P$ to the rotated Hermitian part of A, showing $(C_d)$ is sufficient for invariance and, when $\det(A) \neq 0$, necessary, yielding the strong algebraic constraint $A^d = (−1)^{d−1}\det(A) I$ and symmetry properties of $W(A)$; when $\det(A)=0$, $(C_d)$ forces $W(A)$ to be a disk. The paper develops two equivalent trace-based criteria, analyzes the boundary geometry of $W(A)$ via the tangential representation and Riemann mapping, and specializes to low dimensions: for $d=3$ it fully classifies invariant numerical ranges up to unitary similarity in terms of the model matrix $M(\alpha_1,\alpha_2,\alpha_3)$, while for $d=4$ it shows a rich family of additional solutions beyond the weight-shift form. In particular, the boundary can exhibit curved arcs or straight segments, with line segments arising in both cornered (polygonal) and non-corner cases, and the results provide a framework for identifying invariant numerical ranges across families of matrices, including explicit subfamilies and counterexamples.

Abstract

We characterize the d x d matrices whose numerical ranges are invariant by rotations of angle 2$π$/d.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['th1']}
• Lemma 3
• proof
• Corollary 4
• Lemma 5
• Lemma 6
• proof