557 research outputs found

    The copositive completion problem

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    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

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    Let AA be an element of the copositive cone Cn{\cal C}_n. A zero uu of AA is a nonzero nonnegative vector such that uTAu=0u^TAu = 0. The support of uu is the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the positive entries of uu. A zero uu of AA is called minimal if there does not exist another zero vv of AA 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)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 AA with respect to S+(n)S_+(n) in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone Nn{\cal N}_n of entry-wise nonnegative matrices. For n=5n = 5 matrices which are irreducible with respect to both S+(5)S_+(5) and N5{\cal N}_5 are extremal. For n=6n = 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

    The necessary and sufficient conditions of copositive tensors

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    In this paper, it is proved that (strict) copositivity of a symmetric tensor A\mathcal{A} is equivalent to the fact that every principal sub-tensor of A\mathcal{A} has no a (non-positive) negative H++H^{++}-eigenvalue. The necessary and sufficient conditions are also given in terms of the Z++Z^{++}-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 Rn\mathbb{R}^n.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608

    Extremal copositive matrices with minimal zero supports of cardinality two

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    Let ACnA \in {\cal C}^n be an extremal copositive matrix with unit diagonal. Then the minimal zeros of AA all have supports of cardinality two if and only if the elements of AA are all from the set {1,0,1}\{-1,0,1\}. 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 {1,0,1}\{-1,0,1\} unit diagonal matrices characterized by Hoffman and Pereira in 1973.Comment: 4 page
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