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 $n\ge 2$. In this note we disprove their conjecture by showing that ${\lim }_{n\to \infty }{\theta }_{\max }({\mathcal{C}}_n)=\pi$. 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.

References in corpus (1)

• A characterization of trace zero symmetric nonnegative 5x5 matrices