On the existence of flat orthogonal matrices

TL;DR

This paper explores the existence of flat orthogonal matrices, which are real orthogonal matrices with entries close to 1/√n, as a relaxed version of the Hadamard matrix problem, aiming to understand their potential existence.

Contribution

It introduces the concept of flat orthogonal matrices as a relaxation of Hadamard matrices and investigates their existence, expanding the understanding of orthogonal matrix structures.

Findings

Flat orthogonal matrices can exist for various dimensions.

The relaxation may provide new insights into the Hadamard conjecture.

Connections between flat orthogonal matrices and other matrix classes are discussed.

Abstract

In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices, so the question of existence of flat orthogonal matrices can be viewed as a relaxation of the Hadamard problem.