On the maximum angle between copositive matrices

Abstract

Hiriart-Urruty and Seeger have posed the problem of finding the maximal possible angle $\theta_{\max}(\mathcal{C}_{n})$ between two copositive matrices of order $n$. They have proved that $\theta_{\max}(\mathcal{C}_{2})=\frac{3}{4}\pi$ and conjectured that $\theta_{\max}(\mathcal{C}_{n})$ is equal to $\frac{3}{4}\pi$ for all $n \geq 2$. In this note we disprove their conjecture by showing that $\lim_{n \rightarrow \infty}{\theta_{\max}(\mathcal{C}_{n})}=\pi$. Our proof uses a construction from algebraic graph theory. We also consider the related problem of finding the maximal angle between a nonnegative matrix and a positive semidefinite matrix of the same order.