Almost Hadamard matrices: the case of arbitrary exponents

TL;DR

This paper explores the concept of almost Hadamard matrices, extending the idea to p-norms on orthogonal groups, and discusses theoretical results and open problems in this generalized setting.

Contribution

It introduces a p-norm generalization of almost Hadamard matrices and provides new theoretical insights and open questions in this broader context.

Findings

Reviewed properties of almost Hadamard matrices.

Extended the formalism to p-norms on O(N).

Formulated open problems for future research.

Abstract

In our previous work, we introduced the following relaxation of the Hadamard property: a square matrix $H\in M_N(\mathbb R)$ is called "almost Hadamard" if $U=H/\sqrt{N}$ is orthogonal, and locally maximizes the 1-norm on O(N). We review our previous results, notably with the formulation of a new question, regarding the circulant and symmetric case. We discuss then an extension of the almost Hadamard matrix formalism, by making use of the p-norm on O(N), with $p\in[1,\infty]-{2}$, with a number of theoretical results on the subject, and the formulation of some open problems.