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2021•arXiv (Cornell University)Open access

On Some Quaternionic Hadamard Matrices of Small Order

Logan M. Higginbotham, Chase T. Worley

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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.

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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.

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Available 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.

Key concepts: Complex Hadamard matrix, Hadamard transform, Hadamard's maximal determinant problem, Hadamard three-lines theorem, Hadamard's inequality, Hadamard matrix, Mathematics, Quaternion

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