Related papers: Real Schur norms and Hadamard matrices

Real Schur norms and Hadamard matrices

URL: http://arxiv.org/abs/2206.02863v1
Date: Mon, 6 Jun 2022 19:30:13 GMT
Title: Real Schur norms and Hadamard matrices
Authors: John Holbrook, Nathaniel Johnston, Jean-Pierre Schoch
Abstract summary: We present a preliminary study of Schur norms $|M|_S=max |Mcirc C|: |C|=1$, where M is a matrix whose entries are $pm1$, and $circ$ denotes the entrywise (i.e., Schur or Hadamard) product of the matrix. We show that, if such a matrix M is n-by-n, then its Schur norm is bounded by $sqrtn$, and equality holds if and only
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: 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.

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