Irreducible elements of the copositive cone

Abstract

We call an element A of the n×n copositive cone Cn irreducible with respect to the nonnegative cone Nn if it cannot be written as a nontrivial sum A=C+N of a copositive matrix C and an elementwise nonnegative matrix N (note that our concept of irreducibility differs from the standard one normally studied in matrix theory). This property was studied by Baumert (1965) 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 C5 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 matrix with zero diagonal and a single irreducible matrix.