## 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

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.