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A Note on Approximate Hadamard Matrices

Stefan Steinerberger

Abstract

A Hadamard matrix is a scaled orthogonal matrix with $\pm 1$ entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when $n$ is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when $n > 4$. Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal $0< c < C < \infty$ so that for all $n \geq 1$, there is a matrix $A \in \left\{-1,1\right\}^{n \times n}$ satisfying, for all $x \in \mathbb{R}^n$, $$ c \sqrt{n} \|x\|_2 \leq \|Ax\|_2 \leq C \sqrt{n} \|x\|_2.$$ We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all $n \geq 1$.

A Note on Approximate Hadamard Matrices

Abstract

A Hadamard matrix is a scaled orthogonal matrix with entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when . Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal so that for all , there is a matrix satisfying, for all , We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all .
Paper Structure (9 sections, 2 theorems, 16 equations, 1 figure)

This paper contains 9 sections, 2 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction and Result
  2. Hadamard matrices.
  3. Approximate Hadamard matrices.
  4. Result
  5. Proof
  6. Remarks
  7. Upper bounds.
  8. Lower bounds.
  9. Ultra-flat polynomials

Key Result

Theorem 1

There exist universal $0< c < C < \infty$ such that for all $n \geq 1$, there exists a matrix $A \in \mathbb{R}^{n \times n}$ whose entries are $\pm 1$ such that

Figures (1)

  • Figure 1: Left: $p(e^{it})$ for $1 \leq t \leq 1.05$. Right: a histogram of the values of $| p\left(\cdot\right)|$ when evaluated at the roots of unity.

Theorems & Definitions (5)

  • Theorem : Dong-Rudelson dong
  • Theorem
  • proof
  • Conjecture : Version A: Quantitative Ryser
  • Conjecture : Version B: Ultra-flat at Roots of Unity