557 research outputs found
The copositive completion problem
An n × n real symmetric matrix A is called (strictly) copositive if xTAx ⩾ 0 (\u3e0) whenever x ∈ Rn satisfies x ⩾ 0 (x ⩾ 0 and x ≠ 0). The (strictly) copositive matrix completion problem asks which partial (strictly) copositive matrices have a completion to a (strictly) copositive matrix. We prove that every partial (strictly) copositive matrix has a (strictly) copositive matrix completion and give a lower bound on the values used in the completion. We answer affirmatively an open question whether an n × n copositive matrix A = (aij) with all diagonal entries aii = 1 stays copositive if each off-diagonal entry of A is replaced by min{aij, 1}
Minimal zeros of copositive matrices
Let be an element of the copositive cone . A zero of
is a nonzero nonnegative vector such that . The support of is
the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the
positive entries of . A zero of is called minimal if there does not
exist another zero of 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 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 with
respect to in terms of its minimal zeros. A similar condition is given
for the irreducibility with respect to the cone of entry-wise
nonnegative matrices. For matrices which are irreducible with respect
to both and are extremal. For 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
The necessary and sufficient conditions of copositive tensors
In this paper, it is proved that (strict) copositivity of a symmetric tensor
is equivalent to the fact that every principal sub-tensor of
has no a (non-positive) negative -eigenvalue. The
necessary and sufficient conditions are also given in terms of the
-eigenvalue of the principal sub-tensor of the given tensor. This
presents a method of testing (strict) copositivity of a symmetric tensor by
means of the lower dimensional tensors. Also the equivalent definition of
strictly copositive tensors is given on entire space .Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608
Extremal copositive matrices with minimal zero supports of cardinality two
Let be an extremal copositive matrix with unit diagonal.
Then the minimal zeros of all have supports of cardinality two if and only
if the elements of are all from the set . Thus the extremal
copositive matrices with minimal zero supports of cardinality two are exactly
those matrices which can be obtained by diagonal scaling from the extremal
unit diagonal matrices characterized by Hoffman and Pereira in
1973.Comment: 4 page
- …