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

TL;DR

This paper determines the maximal angle between two copositive matrices of order 3, showing it equals 135 degrees, and characterizes all pairs achieving this maximum.

Contribution

It establishes that the maximal angle for 3x3 copositive matrices is 135 degrees and identifies all matrix pairs that attain this angle.

Findings

Maximal angle between 3x3 copositive matrices is 135 degrees.

Characterization of all matrix pairs achieving the maximal angle.

Extension of known results from 2x2 to 3x3 matrices.

Abstract

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.