On the maximum angle between copositive matrices

TL;DR

This paper disproves a conjecture about the maximum angle between copositive matrices, showing that it approaches π as matrix size increases, using algebraic graph theory techniques.

Contribution

The paper demonstrates that the maximum angle between copositive matrices tends to π, countering previous conjectures, and introduces a novel algebraic graph theory construction.

Findings

Maximum angle between copositive matrices approaches π as n increases

Disproves the conjecture that the maximum angle is always 3/4 π

Uses algebraic graph theory in the proof

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.