# A bridge between quaternionic and complex numerical ranges

## TL;DR

This paper establishes conditions under which quaternionic numerical ranges are convex for complex matrices, showing that real matrices always have convex quaternionic numerical ranges and fully characterizing the case for 2x2 real matrices.

## Contribution

It introduces a sufficient condition linking quaternionic and complex numerical ranges, and characterizes the quaternionic numerical range for 2x2 real matrices.

## Key findings

- Convexity of quaternionic numerical range for complex matrices under certain conditions.
- Real matrices have convex quaternionic numerical range.
- Complete characterization of quaternionic numerical range for 2x2 real matrices.

## Abstract

We obtain a sufficient condition for the convexity of quaternionic numerical range for complex matrices in terms of its complex numerical range. It is also shown that the Bild coincides with complex numerical range for real matrices. From this result we derive that all real matrices have convex quaternionic numerical range. As an example we fully characterize the quaternionic numerical range of $2\times2$ real matrices.