IUMJ

Title: On orthogonal matrices maximizing the 1-norm

Authors: Teodor Banica, Benoit Collins and Jean-Marc Schlenker

Issue: Volume 59 (2010), Issue 3, 839-856

Abstract:

For $U \in O(N)$ we have $\|U\|_1 \leq N\sqrt{N}$, with equality if and only if $U = H/\sqrt{N}$, with $H$ Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1-norm on $O(N)$. The main problem is to compute the $k$-th moment of the 1-norm on $O(N)$, with $k \to \infty$, and we discuss here the general strategy for approaching this problem, with some explicit computations at $k = 1$, $2$ and $N \to \infty$.