Real Schur norms and Hadamard matrices
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929422988214272 |
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| author | Holbrook, John Johnston, Nathaniel Schoch, Jean-Pierre |
| author_facet | Holbrook, John Johnston, Nathaniel Schoch, Jean-Pierre |
| contents | We present a preliminary study of Schur norms $\|M\|_{S}=\max\{ \|M\circ C\|: \|C\|=1\}$, where M is a matrix whose entries are $\pm1$, and $\circ$ denotes the entrywise (i.e., Schur or Hadamard) product of the matrices. We show that, if such a matrix M is n-by-n, then its Schur norm is bounded by $\sqrt{n}$, and equality holds if and only if it is a Hadamard matrix. We develop a numerically efficient method of computing Schur norms, and as an application of our results we present several almost Hadamard matrices that are better than were previously known. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_02863 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Real Schur norms and Hadamard matrices Holbrook, John Johnston, Nathaniel Schoch, Jean-Pierre Combinatorics Quantum Physics We present a preliminary study of Schur norms $\|M\|_{S}=\max\{ \|M\circ C\|: \|C\|=1\}$, where M is a matrix whose entries are $\pm1$, and $\circ$ denotes the entrywise (i.e., Schur or Hadamard) product of the matrices. We show that, if such a matrix M is n-by-n, then its Schur norm is bounded by $\sqrt{n}$, and equality holds if and only if it is a Hadamard matrix. We develop a numerically efficient method of computing Schur norms, and as an application of our results we present several almost Hadamard matrices that are better than were previously known. |
| title | Real Schur norms and Hadamard matrices |
| topic | Combinatorics Quantum Physics |
| url | https://arxiv.org/abs/2206.02863 |