On determinants and eigenvalue theory of tensors

Hu, Shenglong; Huang, Zheng-Hai; Ling, Chen; Qi, Liqun

doi:10.1016/j.jsc.2012.10.001

Cited by 141 publications

(127 citation statements)

References 24 publications

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“…The singularity of a square matrix can also be checked by its spectrum, that is, A is singular if and only if A has zero as one of its eigenvalues. This also applies to the high order tensors due to Hu [7] and the recent work by Shao [14] where it is shown that the determinant of a cubic tensor can be written as a product of some powers of eigenvalues of A. There are many definitions of singularities of tensors.…”

Section: The Singularity Of Vandermonde Tensorsmentioning

confidence: 94%

Hankel tensors, Vandermonde tensors and their positivities

2016

Linear Algebra and its Applications

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Section: The Singularity Of Vandermonde Tensorsmentioning

confidence: 94%

Hankel tensors, Vandermonde tensors and their positivities

2016

Linear Algebra and its Applications

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“…Chang et al [4] defined the geometric multiplicity of an eigenvalue λ, meanwhile, Hu et al [17] considered the algebraic multiplicity of an eigenvalue λ. Similarly, we can define the geometric and algebraic multiplicity of an Z-eigenvalue.…”

Section: Downloaded By [University Of Nebraska Lincoln] At 18:17 01 mentioning

confidence: 99%

Perturbation bounds of tensor eigenvalue and singular value problems with even order

Che

Wei

2015

Linear and Multilinear Algebra

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The main purpose of this paper is to investigate the perturbation bounds of the tensor eigenvalue and singular value problems with even order. We extend classical definitions from matrices to tensors, such as, λ-tensor and the tensor polynomial eigenvalue problem. We design a method for obtaining a mode-symmetric embedding from a general tensor. For a given tensor, if the tensor is mode-symmetric, then we derive perturbation bounds on an algebraic simple eigenvalue and Z-eigenvalue. Otherwise, based on symmetric or modesymmetric embedding, perturbation bounds of an algebraic simple singular value are presented. For a given tensor tuple, if all tensors in this tuple are modesymmetric, based on the definition of a λ-tensor, we estimate perturbation bounds of an algebraic simple polynomial eigenvalue. In particular, we focus on tensor generalized eigenvalue problems and tensor quadratic eigenvalue problems.

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“…Very recently, Hu et al [6] investigated the characteristic polynomial of a tensor through the determinant and the higher order traces, which can be used to compute the eigenvalues of tensors. However it is not easy to compute all eigenvalues (or H-eigenvalues) of a tensor when its order is very large.…”

Section: Introductionmentioning

confidence: 99%

“…However it is not easy to compute all eigenvalues (or H-eigenvalues) of a tensor when its order is very large. So it is difficult in some cases to identify the positive definiteness of a tensor by the method provided in [6].…”

Section: Introductionmentioning

confidence: 99%

A new eigenvalue inclusion set for tensors and its applications

Chen

2015

Linear Algebra and its Applications

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A new tensor eigenvalue inclusion set is given, and proved to be tighter than those in L.Q. Qi (2005) [18] and C.Q. Li, Y.T. Li, X. Kong (2014) [12]. In addition, we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by Qi (2005) for weakly irreducible tensors. As applications, we give new bounds for the spectral radius of nonnegative tensors, some sufficient conditions for a tensor to be a strong M -tensor and some inequalities to identify the positive definiteness for an even-order real supersymmetric tensor.

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On determinants and eigenvalue theory of tensors

Cited by 141 publications

References 24 publications

Hankel tensors, Vandermonde tensors and their positivities

Hankel tensors, Vandermonde tensors and their positivities

Perturbation bounds of tensor eigenvalue and singular value problems with even order

A new eigenvalue inclusion set for tensors and its applications

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