On some 4-by-4 matrices with bi-elliptical numerical ranges

TL;DR

This paper characterizes 4-by-4 matrices with scalar diagonal blocks whose numerical ranges are the convex hull of two non-concentric ellipses, providing explicit descriptions and special cases, with applications to reciprocal matrices.

Contribution

It offers a complete description of certain 4-by-4 matrices with bi-elliptical numerical ranges, including simplified forms in special cases and an application to reciprocal matrices.

Findings

Characterization of 4-by-4 matrices with bi-elliptical numerical ranges.

Explicit descriptions for special cases where differences are real or imaginary.

Application to reciprocal matrices.

Abstract

A complete description of 4-by-4 matrices $\begin{bmatrix}\alpha I & C \\D & \beta I\end{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^*$ also is a scalar multiple of the identity. In particular cases when in addition $\alpha-\beta$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.