Real Schur norms and Hadamard matrices

Abstract:

We present a preliminary study of Schur norms \|M\|_{\textup{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\times 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.

Authors:

  • John Holbrook
  • Nathaniel Johnston
  • Jean-Pierre Schoch

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Cite as:

  • J. Holbrook, N. Johnston, and J.-P. Schoch. Real Schur norms and Hadamard matrices. E-print: arXiv:2206.02863 [math.CO], 2022.

Supplementary material:

  • EquivalenceClasses.zip – MATLAB code for finding all equivalence classes of (+1,-1) matrices of a given size, as well as a MATLAB file containing representatives of all equivalence classes of size up to 7×7
  • EquivClasses.txt – A summary of all equivalence classes of (+1,-1) matrices of size 6×6 or less. For the equivalence classes in the 7×7 case, load the MATLAB file above instead.
  • SchurNorm.zip – MATLAB code for computing the Schur norm of a matrix, and for finding the largest Schur norm of a (circulant or not) (+1,-1) matrix of a given size.
  • largeschurnorm.jl – Julia code by Jean-Pierre Schoch for computing the Schur norm of a matrix, and for finding the largest Schur norm of a (circulant or not) (+1,-1) matrix of a given size.
  • circulatschurnorms.jl – Julia code by Jean-Pierre Schoch for computing the largest Schur norm of a circulant (+1,-1) matrix of a given size.
  • MaximalSchurNorms.txt – A text file containing the circulant and non-circulant (+1,-1) matrices with largest Schur norm that we have been able to find, for sizes up to 24×24. The circulants are all known to be optimal, and the non-circulants are known to be optimal up to 8×8.
  • OptimalOrthogonalL1Norm.txt – A text file containing an orthogonal matrix with largest entrywise 1-norm that we have been able to find, for sizes from 3×3 to 24×24. The matrices up to size 8×8 are known to be optimal.