Irreducible elements of the copositive cone

An element A of the n× n copositive cone C is called irreducible with respect to the nonnegative cone 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 [2] 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 5×5 copositive matrices we give a complete characterisation of all irreducible matrices. We show that those irreducible matrices in C which are not positive semidefinite can be parameterized in a semi-trigonometric way. Finally, we prove that every 5×5 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.