Minimal zeros of copositive matrices

Abstract

Let A be an element of the copositive cone Cn. A zero u of A is a nonzero nonnegative vector such that uT Au = 0. The support of u is the index set supp u c {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) 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) in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone Nn of entry-wise nonnegative matrices. For n = 5 matrices which are irreducible respect to both S+(5) and N5 are extremal. For n = 6 a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.