Real Schur norms and Hadamard matrices (Q6608147)

This interesting paper presents a preliminary study about the quantity \[\max_{A,B} \{\|B\|/\|A\| : |b_{ij}| \leq |a_{ij}| \text{ for all } i,j\},\] denoted by \(c_n\) when \(A,B\in M_n(\mathbb{C})\) and \(r_n\) when \(A,B\in M_n(\mathbb{R})\), where \(\|\cdot\|\) is the usual operator norm. This natural question, stemming from the observation that reducing some entries of a matrix can sometimes increase its norm, was studied back in 1911 by \textit{I. Schur} [J. Reine Angew. Math. 140, 1--28 (1911; JFM 42.0367.01)]. Moreover, it is known that (1) \(c_n=\sqrt{n}\), (2) \(r_n\leq \sqrt{n}\) and (3) \(r_n = \sqrt{n}\) if and only if there exists an \(n\times n\) Hadamard matrix, see [\textit{E. C. Johnsen}, Linear Multilinear Algebra 4, 277--279 (1977; Zbl 0344.15011)]. \N\NThe authors show, among other things, that \(r_n\) cannot be far off from \(\sqrt{n}\): If \(n\in\mathbb{N}\) and \(p\) is a prime number satisfying \(p\leq n/2-1\), then \(r_n\geq \sqrt{2(p+1)}\). Moreover, the authors develop a method that allows numerical approximation to any accuracy of \(r_n\) via semidefinite programming, which is therefore in polynomial time (for individual matrices). They then use this method to approximate \(r_n\) for \(n\leq 24\), and to compute it exactly for \(n\leq 8\).