On Some Quaternionic Hadamard Matrices of Small Order

TL;DR

This paper introduces and classifies quaternionic Hadamard matrices of small orders, exploring their properties, types, and construction methods, revealing a richer structure than complex Hadamard matrices.

Contribution

It provides the first classification of quaternionic Hadamard matrices with circulant core of orders 2 to 5 and introduces quaternionic Hadamard matrices of Butson type.

Findings

Quaternionic Hadamard matrices are more diverse than complex ones.

Classification of quaternionic Hadamard matrices of orders 2-5.

Methods to construct quaternionic Hadamard matrices from real and complex matrices.

Abstract

We introduce Hadamard matrices whose entries are quaternionic. We then go on to provide classification of quaternionic Hadamard matrices of circulant core of orders 2 through 5. We also introduce quaternionic Hadamard matrices of Butson type and ways to create quaternionic Hadamard matrices from real and complex Hadamard matrices. Examples are shown that showcase how Hadamard matrices over the quaternions are richer than Hadamard matrices over the complex numbers.