Real Schur norms and Hadamard matrices

Abstract

We present a preliminary study of Schur norms $\Vert M{\Vert }_S=\max \{\Vert M\circ C\Vert :\Vert C\Vert =1\}$, where M is a matrix whose entries are $\pm 1$, 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.

17 pages

References in corpus (2)

• Simpler semidefinite programs for completely bounded norms
• Real Schur norms and Hadamard matrices

Cited by in corpus (2)

• Real Schur norms and Hadamard matrices
• Unique Matrix Factorizations associated to Bilinear Forms and Schur Multipliers