On the convexity and circularity of the numerical range of nilpotent quaternionic matrices

TL;DR

This paper investigates conditions under which the numerical range of nilpotent quaternionic matrices is circular or convex, providing new criteria and characterizations for these geometric properties.

Contribution

It introduces new sufficient conditions for circularity and convexity of the numerical range of nilpotent matrices, especially for 3x3 cases, linking graph cycles to these properties.

Findings

Numerical range of nilpotent matrices can be circular under certain graph cycle conditions.

Adding a diagonal real matrix to a nilpotent matrix results in a convex numerical range.

Complete characterization for 3x3 nilpotent matrices' numerical range properties.

Abstract

We provide a sufficient condition for the numerical range of a nilpotent matrix N to be circular in terms of the existence of cycles in an undirected graph associated with N. We prove that if we add to this matrix N a diagonal real matrix D, the matrix D + N has convex numerical range. For 3 x 3 nilpotent matrices, we strength further our results and obtain necessary and sufficient conditions for circularity and convexity of the numerical range.