Real Schur norms and Hadamard matrices

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 bounde...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Holbrook, John, Johnston, Nathaniel, Schoch, Jean-Pierre
Format: Article
Language:English
Subjects:
Norms
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2206.02863