On the maximal angle between copositive matrices

Abstract

Hiriart-Urruty and Seeger have posed the problem of finding the maximal possible angle θmax(Cn) between two copositive matrices of order n. They have proved that θmax(C 2 ) = 3 4 π and conjectured that θmax(Cn) is equal to 3 4 π for all n ≥ 2. In this note we disprove their conjecture by showing that limn→∞ θmax(Cn) = 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.

When a maximal angle among cones is nonobtuse

“…What is the maximal angle that can be formed by two copositive matrices? Hiriart-Urruty and Seeger (2010) conjecture that the answer is 3π/4, but Goldberg and Shaked-Monderer (2014) show that the maximal angle tends towards π as n grows large.…”

The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices

“…They further conjectured that the maximal angle for two n × n copositive matrices is also 3π 4 for n ≥ 3. In addressing this conjecture, Goldberg and Shaked-Monderer [3] constructed a sequences of pairs (P k , N k ) such that the angle between P k and N k approaches π as n k → ∞, where P k is an n k × n k positive semidefinite matrix and N k is an n k × n k nonnegative matrix. Since the set of copositive matrices contains positive semidefinite matrices and nonnegative matrices, the construction of such a sequence disproved the conjecture of Hiriart-Urruty and Seeger.…”

“…Goldberg and Shaked-Monderer pointed out in [3] that the problem of calculating or estimating the maximal angle between an n × n positive semidefinite matrix and an n × n nonnegative matrix is interesting in its own right. In [3], Goldberg and Shaked-Monderer studied the maximal angle between an n × n semidefinite matrix and an n × n nonnegative matrix for n = 3, 4, and 5. They proved that for n ≤ 4, the maximal angle between an n × n positive semidefinite matrix and an n × n nonnegative matrix is 3π 4 .…”

“…They proved that for n ≤ 4, the maximal angle between an n × n positive semidefinite matrix and an n × n nonnegative matrix is 3π 4 . For n = 5, a nonnegative matrix and a semidefinite matrix are constructed in [3] to show that the maximal angle is strictly greater than 3π 4 . In the proofs of the results for n ≤ 4, an optimization problem involving a convex function was used in [3].…”

Open problems in the theory of completely positive and copositive matrices

“…6.13] it was proved that θ max (COP 2 ) = 3π/4, and it was conjectured that θ max (COP n ) = 3π/4 for all n ≥ 2. However, this was disproved in [30], where it The proof is based on constructing sequences of matrices P k ∈ PSD n k and N k ∈ N n k of increasing order with lim k→∞ arccos P k , N k = π. Since P k , N k ∈ COP n k and COP n is pointed for all n, this implies (3.1).…”