Minimal zeros of copositive matrices

Let A   be an element of the copositive cone CnCn. A zero u of A   is a nonzero nonnegative vector such that uTAu=0uTAu=0. The support of u   is the index set suppu⊂{1,…,n} corresponding to the positive entries of u. A zero u of A is called minimal if there does not exist another zero v of A such that its support supp v is a strict subset of supp u  . We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone S+(n)S+(n) of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix A   with respect to S+(n)S+(n) in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone NnNn of entry-wise nonnegative matrices. For n=5n=5 matrices which are irreducible with respect to both S+(5)S+(5) and N5N5 are extremal. For n=6n=6 a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.