Invitation to Hadamard matrices

TL;DR

This paper explores the theory of Hadamard matrices, both real and complex, highlighting their properties, examples like Fourier matrices, and their geometric and analytic aspects.

Contribution

It provides an overview of the basic theory of complex Hadamard matrices, emphasizing their structure, examples, and mathematical significance.

Findings

Complex Hadamard matrices generalize Fourier matrices.

They have applications in geometry and analysis.

The paper discusses their properties and examples.

Abstract

An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main examples are the Fourier matrices, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$, and at the level of the general theory, the complex Hadamard matrices can be thought of as being some sort of exotic, generalized Fourier matrices. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.