On orthogonal matrices maximizing the 1-norm

TL;DR

This paper investigates the maximum 1-norm of orthogonal matrices, establishing conditions for equality and exploring algebraic and analytic methods to compute the norm's moments, especially as the moment order grows large.

Contribution

It provides a detailed analysis of the algebraic and analytic aspects of maximizing the 1-norm on orthogonal matrices, including moments computation.

Findings

Maximum 1-norm achieved by scaled Hadamard matrices

Characterization of matrices attaining the maximum 1-norm

Discussion on the asymptotic behavior of the 1-norm moments

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, with $k\to\infty$, and we present a number of general comments in this direction.