# The star-center of the quaternionic numerical range

## TL;DR

This paper proves that the quaternionic numerical range is always star-shaped, identifies its star-center, and describes its geometric structure using tangents to the lower bild, advancing understanding of quaternionic matrix analysis.

## Contribution

It establishes the star-shaped property of the quaternionic numerical range and characterizes its star-center through geometric analysis, a novel contribution in quaternionic matrix theory.

## Key findings

- Quaternionic numerical range is always star-shaped.
- Star-center determined by equivalence classes of the bild's star-center.
- Geometric shape of the star-center's upper part is defined by two tangent lines.

## Abstract

In this paper we prove that the quaternionic numerical range is always star-shaped and its star-center is given by the equivalence classes of the star-center of the bild. We determine the star-center of the bild, and consequently of the numerical range, by showing that the geometrical shape of the upper part of the center is defined by two lines, tangents to the lower bild.