Minimal zeros of copositive matrices

Abstract

Let $A$ be an element of the copositive cone $C_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu=0$. The support of $u$ is the index set $supp u \subset \{1,…,n\}$ 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 $N_n$ of entry-wise nonnegative matrices. For $n=5$ matrices which are irreducible with respect to both $S_+(5)$ and $N_5$ are extremal. For $n=6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided