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2013•Linear and Multilinear AlgebraOpen access

Necessary and sufficient conditions for copositive tensors

Yisheng Song, Liqun Qi

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Abstract

In this paper, it is proved that a symmetric tensor is (strictly) copositive if and only if each of its principal sub-tensors has no (non-positive) negative -eigenvalue. Necessary and sufficient conditions for (strict) copositivity of a symmetric tensor are also given in terms of -eigenvalues of the principal sub-tensors of that tensor. This presents a method for testing (strict) copositivity of a symmetric tensor by means of lower dimensional tensors. Also, an equivalent definition of strictly copositive tensors is given on the entire space .

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In this paper, it is proved that a symmetric tensor is (strictly) copositive if and only if each of its principal sub-tensors has no (non-positive) negative -eigenvalue. Necessary and sufficient conditions for (strict) copositivity of a symmetric tensor are also given in terms of -eigenvalues of the principal sub-tensors of that tensor. This presents a method for testing (strict) copositivity of a symmetric tensor by means of lower dimensional tensors. Also, an equivalent definition of strictly copositive tensors is given on the entire space .

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Available abstract

In this paper, it is proved that a symmetric tensor is (strictly) copositive if and only if each of its principal sub-tensors has no (non-positive) negative -eigenvalue. Necessary and sufficient conditions for (strict) copositivity of a symmetric tensor are also given in terms of -eigenvalues of the principal sub-tensors of that tensor. This presents a method for testing (strict) copositivity of a symmetric tensor by means of lower dimensional tensors. Also, an equivalent definition of strictly copositive tensors is given on the entire space .

Key concepts: Tensor (intrinsic definition), Constructive, Principal (computer security), Eigenvalues and eigenvectors, Symmetric tensor, Mathematics, Mathematical economics, Pure mathematics

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