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

Daniel Gourion

TL;DR

This work resolves the maximal angle between copositive $3\times3$ matrices by proving $\theta_3 = \frac{3\pi}{4}$ and providing a complete description of all antipodal pairs achieving this bound (up to symmetry and scaling). Through a finite case analysis over sign patterns of off-diagonal entries and a combination of linear-algebraic and optimization techniques, the authors show that all other configurations yield angles at most $\frac{3\pi}{4}$, with several cases attaining the bound only in a limiting sense. The result sharpens understanding of the geometry of the copositive cone $\mathcal{C}_3$ within the space of symmetric matrices, and highlights the intricate structure that emerges even at moderate dimension. The conclusions contribute to the broader study of maximal angles between convex cones and have implications for optimization contexts where copositive matrices arise.

Abstract

In 2010, Hiriart-Urruty and Seeger posed the problem of finding the maximal possible angle $θ_n$ between two copositive matrices of order $n$. They proved that $θ_2=\frac{3}{4}π$. In this paper, we study the maximal angle between two copositive matrices of order 3. We show that $θ_3=\frac{3}{4}π$ 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.

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

TL;DR

This work resolves the maximal angle between copositive matrices by proving and providing a complete description of all antipodal pairs achieving this bound (up to symmetry and scaling). Through a finite case analysis over sign patterns of off-diagonal entries and a combination of linear-algebraic and optimization techniques, the authors show that all other configurations yield angles at most , with several cases attaining the bound only in a limiting sense. The result sharpens understanding of the geometry of the copositive cone within the space of symmetric matrices, and highlights the intricate structure that emerges even at moderate dimension. The conclusions contribute to the broader study of maximal angles between convex cones and have implications for optimization contexts where copositive matrices arise.

Abstract

In 2010, Hiriart-Urruty and Seeger posed the problem of finding the maximal possible angle between two copositive matrices of order . They proved that . In this paper, we study the maximal angle between two copositive matrices of order 3. We show that 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.
Paper Structure (12 sections, 15 theorems, 45 equations)

This paper contains 12 sections, 15 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Copositive matrices
  4. Maximal angle
  5. The maximal angle between copositive matrices of size 3
  6. Case $(0,0)$
  7. Case $(1,0)$
  8. Case $(3,0)$
  9. Case $(2,0)$
  10. Case $(1,1)$
  11. Case $(2,1)$
  12. Concluding remark

Key Result

Proposition 2.1

Let $A=(a_{ij})_{1\leq 1,j\leq n}$ be a copositive matrix of size $n$ then we have:

Theorems & Definitions (26)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Lemma 2.7
  • ...and 16 more