Symmetric nonnegative tensors and copositive tensors

By Liqun Qi. 2012

290 citations recorded by the source corpus.

Abstract We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.

Subfields: Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3347cec00ec6505d12aa335281cb2b51115bd9a4

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In Computational Mathematics

In Computational Theory and Mathematics

Citations in the corpus

H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors (2013, methodology): “Thus, we may use the recently developed theory and algorithms on eigenvalues of nonnegative tensors [4, 5, 8, 9, 17, 18, 22, 27, 28] to study L and Q.”
Controlling Recurrent Neural Networks by Conceptors (2014, background): “This may turn out to be a challenge because a mathematical theory of “semi positive-definite tensors” seems to be only in its infancy (compare [84]).”
Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition (2013, background): “✷ As a positive semi-definite symmetric tensor is copositive [11], the condition v(i−1)m ≥ 0 for i ∈ [n] is also a necessary condition for an even order Hankel tensor to be positive semi-definite.”
Infinite and finite dimensional Hilbert tensors (2014, background): “Since Hilbert tensor is positive (all entries are positive) and symmetric, then the following conclusions (i) were easily obtained from Chang, Pearson and Zhang [2], Qi [12], Song and Qi [15] and Yang and Yang [18, 17], and the conclusions (ii) can be obtained from Chang, Pearson and Zhang [1] and S”
M-Tensors and Some Applications (2014, unknown): “Some existing results on symmetric Z-tensors [23] are summarized in the following lemma.”
Open problems in the theory of completely positive and copositive matrices (2015, background): “In [53] it was shown that any symmetric tensor whose off-diagonal entries are nonpositive is copositive if and only if it is positive semidefinite.”
The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph (2013, background): “Please refer to [15,9,5,18,16,13,22,21,24,25,10,3,8,14,17, 19,23,26] for recent developments on spectral hypergraph theory and the essential tools from spectral theory of nonnegative tensors.”
Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors (2014, background): “The concept of (strictly) copositive tensors was first introduced by Qi in [21].”
Vacuum stability of a general scalar potential of a few fields (2016, background): “Copositive tensors are defined in obvious analogy to copositive matrices [46]: A real symmetric tensor A of order m and dimension n is copositive if Ax > 0 for all x ∈ R+.”