Minimal zeros of copositive matrices

Abstract

Let $A$ be an element of the copositive cone ${\mathcal{C}}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^{T}Au=0$. The support of $u$ is the index set $\mbox{supp}u \subset \{1,\dots,n\}$ 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 $\mbox{supp}v$ is a strict subset of $\mbox{supp}u$. 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 ${\mathcal{N}}_n$ of entry-wise nonnegative matrices. For $n=5$ matrices which are irreducible with respect to both $S_{+}(5)$ and ${\mathcal{N}}_5$ are extremal. For $n=6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.

Some conditions and proofs simplified