Real Schur norms and Hadamard matrices

We present a preliminary study of Schur norms parallel to M parallel to(S) = max{parallel to M omicron C parallel to : parallel to C parallel to = 1}, where M is a matrix whose entries are +/- 1, and omicron 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 n x n, then its Schur norm is bounded by root 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.