# Irreducible elements of the copositive cone

An element $A$ of the $n \times n$ copositive cone $\copos{n}$ is called irreducible with respect to the nonnegative cone~$\NNM{n}$ if it cannot be written as a nontrivial sum $A = C+N$ of a copositive matrix $C$ and an elementwise nonnegative matrix $N$. This property was studied by Baumert~\cite{Baumert65} who gave a characterisation of irreducible matrices. We demonstrate here that Baumert's characterisation is incorrect and give a correct version of his theorem which establishes a necessary and sufficient condition for a copositive matrix to be irreducible. For the case of $5\times 5$ copositive matrices we give a complete characterisation of all irreducible matrices. We show that those irreducible matrices in $\copos{5}$ which are not positive semidefinite can be parameterized in a semi-trigonometric way. Finally, we prove that every $5 \times 5$ copositive matrix which is not the sum of a nonnegative and a semidefinite matrix can be expressed as the sum of a nonnegative and a single irreducible matrix.

Preprint, 2012