- Irreducible elements of the copositive cone Peter J.C. Dickinson (peter.dickinsoncantab.net) Mirjam D�r (dueruni-trier.de) Luuk Gijben (L.Gijbenrug.nl) Roland Hildebrand (roland.hildebrandimag.fr) Abstract: 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. Keywords: $5 \times 5$ copositive matrix, irreducibility, exceptional copositive matrix Category 1: Linear, Cone and Semidefinite Programming (Other ) Citation: Preprint, 2012 Download: [PDF]Entry Submitted: 03/08/2012Entry Accepted: 03/08/2012Entry Last Modified: 03/28/2017Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.