## Real Schur norms and Hadamard matrices

**Abstract:**

We present a preliminary study of Schur norms , where M is a matrix whose entries are , and denotes the entrywise (i.e., Schur or Hadamard) product of the matrices. We recover a result of Johnsen that says that, if such a matrix M is , then its Schur norm is bounded by , 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

**Download:**

- Official publication from Linear and Multilinear Algebra
- Preprint from arXiv:2206.02863 [math.CO]
- Local preprint

**Cite as:**

- J. Holbrook, N. Johnston, and J.-P. Schoch. Real Schur norms and Hadamard matrices.
*Linear and Multilinear Algebra*, 2023. doi:10.1080/03081087.2023.2212317

**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.