Asymptotically optimal approximate Hadamard matrices

TL;DR

This paper investigates approximate Hadamard matrices, demonstrating that their condition number approaches 1 as size increases and providing explicit families of such matrices.

Contribution

It proves the asymptotic optimality of approximate Hadamard matrices and identifies explicit infinite families of these matrices.

Findings

Condition number approaches 1 as n increases

Explicit infinite families of approximate Hadamard matrices identified

Well-conditioned matrices with entries in {±1}

Abstract

In this paper, we study approximate Hadamard matrices, that is, well-conditioned $n\times n$ matrices with all entries in $\{\pm1\}$. We show that the smallest-possible condition number goes to $1$ as $n\to\infty$, and we identify some explicit infinite families of approximate Hadamard matrices.