Non-parallel Flat Portions on the Boundaries of Numerical Ranges of 4-by-4 Nilpotent Matrices

TL;DR

This paper characterizes 4-by-4 nilpotent matrices with numerical ranges featuring non-parallel flat boundary portions, revealing their symmetry and the range of angles between these flat segments.

Contribution

It provides a complete characterization of such matrices and their numerical ranges, including symmetry properties and possible angles between flat portions.

Findings

Numerical ranges are symmetric about a line through the origin.

All angles between flat portions are attainable.

Flat portions are on lines equidistant from the origin.

Abstract

The 4-by-4 nilpotent matrices the numerical ranges of which have non-parallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.