On the maximal angle between copositive matrices

Hiriart-Urruty and Seeger have posed the problem of finding the maximal possible angle ?_{max}(C_n) between two copositive matrices of order n [J.-B. Hiriart-Urruty and A. Seeger. A variational approach to copositive matrices. SIAM Rev., 52:593–629, 2010.]. They have proved that ?_{max}(C_2) = (3/4)pi and conjectured that ?_{max}(C_n) is equal to (3/4)pi for all n ? 2. In this note, their conjecture is disproven by showing that lim_{n??} ?_{max}(C_n) = pi. The proof uses a construction from algebraic graph theory. The related problem of finding the maximal angle between a nonnegative matrix and a positive semidefinite matrix of the same order is considered in this paper.