A Note on Approximate Hadamard Matrices
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910337690763264 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | 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 $n \geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13202 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note on Approximate Hadamard Matrices Steinerberger, Stefan Combinatorics Functional Analysis 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 $n \geq 1$. |
| title | A Note on Approximate Hadamard Matrices |
| topic | Combinatorics Functional Analysis |
| url | https://arxiv.org/abs/2402.13202 |