The maximal angle between $3 \times 3$ copositive matrices

In 2010, Hiriart-Urruty and Seeger posed the problem of finding the maximal possible angle $\theta_n$ between two copositive matrices of order $n$. They proved that $\theta_2=\frac{3}{4}\pi$. In this paper, we study the maximal angle between two copositive matrices of order 3. We show that $\theta_3=\frac{3}{4}\pi$ and give all possible pairs of matrices achieving this maximal angle. The proof is based on case analysis and uses optimization and basic linear algebra techniques.