The angular spectrum of the $\boldsymbol{3\times 3}$ copositive cone

Given a vector $x$ and a closed convex cone $\mathcal{C}$ in an $n$-dimensional inner product space. If $x$ is not in the dual cone of $\mathcal{C}$, then the maximal angle between $x$ and $\mathcal{C}$ is greater than $\frac{\pi}{2}$. In this case, a formula regarding the maximal angle between $x$ and $\mathcal{C}$ is given in terms of the metric projection of $-x$ on $\mathcal{C}$. Critical angles between two convex cones that are greater than or equal to $\frac{\pi}{2}$ are shown to be Nash angles by using this formula. Furthermore, some properties of critical pairs of the cone that is the sum of the $n\times n$ positive semidefinite cone and the cone of all $n\times n$ symmetric nonnegative matrices are presented. Since the $n\times n$ copositive cone is the same as the sum of the $n\times n$ positive semidefinite cone and the cone of all $n\times n$ symmetric nonnegative matrices for $n\le 4$, a detailed discussion on how to obtain the angular spectrum of the copositive cone of order 3 is given using the results proved in this paper.