Complex Hadamard matrices from Sylvester inverse orthogonal matrices

TL;DR

This paper introduces a new method for parametrizing complex inverse orthogonal matrices, generalizing complex Hadamard matrices, and provides new parametrizations for specific dimensions when parameters are on the unit circle.

Contribution

It presents a novel approach to generate complex Hadamard matrices from inverse orthogonal matrices by doubling the size of inverse complex conference matrices.

Findings

New parametrizations for complex Hadamard matrices in dimensions 8, 10, 12.

Method generalizes previous constructions of Hadamard matrices.

Matrices depend on non-zero complex parameters, with special cases on the unit circle.

Abstract

A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on non zero complex parameters. The method we use is via doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into complex Hadamard matrices, and in this way we find new parametrizations of Hadamard matrices for dimensions n= 8,10,12.