Real eigenvalues of nonsymmetric tensors

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Abstract

This paper discusses the computation of real \(\mathtt {Z}\)-eigenvalues and \(\mathtt {H}\)-eigenvalues of nonsymmetric tensors. A generic nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. The number of \(\mathtt {H}\)-eigenvalues is finite for all tensors. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every tensor that has finitely many real \(\mathtt {Z}\)-eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every tensor, we can compute all its real \(\mathtt {H}\)-eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations.

Keywords

Tensor \(\mathtt {Z}\)-eigenvalue \(\mathtt {H}\)-eigenvalue Lasserre’s hierarchy Semidefinite relaxation 

Mathematics Subject Classification

15A18 15A69 90C22 

Notes

Acknowledgements

Jiawang Nie was partially supported by the NSF Grants DMS-1417985 and DMS-1619973. Xinzhen Zhang was partially supported by the National Natural Science Foundation of China (Grant No. 11471242).

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA
  2. 2.School of MathematicsTianjin UniversityTianjinChina

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