On the maximum angle between copositive matrices

Hiriart-Urruty and Seeger have posed the problem of finding the maximal possible angle $θ_{\max}(\mathcal{C}_{n})$ between two copositive matrices of order $n$. They have proved that $θ_{\max}(\mathcal{C}_{2})=\frac{3}{4}π$ and conjectured that $θ_{\max}(\mathcal{C}_{n})$ is equal to $\frac{3}{4}π$ for all $n \geq 2$. In this note we disprove their conjecture by showing that $\lim_{n \rightarrow \infty}{θ_{\max}(\mathcal{C}_{n})}=π$. 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.

No reviews yet.

No comments yet.