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Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
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
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