Let A be an element of the copositive cone Cn. A zero u of A
is a nonzero nonnegative vector such that uTAu=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 Nn of entry-wise
nonnegative matrices. For n=5 matrices which are irreducible with 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.Comment: Some conditions and proofs simplifie