A note on approximate Hadamard matrices

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-06-04 DOI:10.1007/s10623-024-01430-w

Stefan Steinerberger

引用次数: 0

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 $n \ge 1$, there is a matrix $A \in \left\{ -1,1\right\} ^{n \times n}$ satisfying, for all $x \in \mathbb {R}^n$,

$$\begin{aligned} c \sqrt{n} \Vert x\Vert _2 \le \Vert Ax\Vert _2 \le C \sqrt{n} \Vert x\Vert _2. \end{aligned}$$

We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all $n \ge 1$.

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关于近似哈达玛矩阵的说明

哈达玛矩阵是一个具有 $\pm 1$ 条目的按比例正交矩阵。这样的矩阵存在于某些维度中：哈达玛猜想是，当 n 是 4 的倍数时，这样的矩阵总是存在的。雷塞尔提出的一个猜想是，当 $n > 4$ 时，不存在环形哈达玛矩阵。最近，Dong 和 Rudelson 证明了所有维度上近似 Hadamard 矩阵的存在：存在普遍的（0< c< C<）矩阵，这样对于所有的（n），都有一个矩阵（A）满足，对于所有的（x），$$\begin{aligned} c \sqrt{n}\Vert x\Vert _2 \le \Vert Ax\Vert _2 \le C \sqrt{n}\Vert x\Vert _2.\end{aligned}$$我们注意到，由于平利特尔伍德多项式的存在，对于所有的 $n \ge 1$ 都存在环形近似哈达玛矩阵。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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