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Acta Applicandae Mathematicae

, Volume 162, Issue 1, pp 165–183 | Cite as

Eigenvectors of Tensors—A Primer

  • Sebastian WalcherEmail author
Article

Abstract

We give an introduction to the theory and to some applications of eigenvectors of tensors (in other words, invariant one-dimensional subspaces of homogeneous polynomial maps), including a review of some concepts that are useful for their discussion. The intent is to give practitioners an overview of fundamental notions, results and techniques.

Keywords

Bezout theorem Brouwer degree Liquid crystal 

Mathematics Subject Classification (2010)

15A69, 14Q05 34A05, 76A15 

References

  1. 1.
    Börnsen, J.-P., van de Ven, A.E.M.: Tangent developable orbit space of an octupole. Preprint (2018). arXiv:1807.04817
  2. 2.
    Cartwright, D., Sturmfels, B.: The number of eigenvalues of a tensor. Linear Algebra Appl. 438, 942–952 (2013) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chen, Y., Qi, L., Virga, E.G.: Octupolar tensors for liquid crystals. J. Phys. A 51(2), 025206 (2018) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cox, D.A., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, New York (2004) zbMATHGoogle Scholar
  5. 5.
    Decker, W., Lossen, Ch.: Computing in Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 16. Springer, Berlin (2006) zbMATHGoogle Scholar
  6. 6.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) zbMATHGoogle Scholar
  7. 7.
    Dumortier, F., Llibre, J., Artes, J.: Qualitative Theory of Planar Differential Systems. Springer, Berlin (2006) zbMATHGoogle Scholar
  8. 8.
    Gaeta, G., Virga, E.G.: Octupolar order in three dimensions. Eur. Phys. J. E 39, 113 (2016) Google Scholar
  9. 9.
    Gantmacher, F.R.: Applications of the Theory of Matrices. Dover, Mineola (2005) Google Scholar
  10. 10.
    Kaplan, J.L., Yorke, J.A.: Nonassociative, real algebras and quadratic differential equations. Nonlinear Anal. 3, 49–51 (1977) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Milnor, J.W.: Topology from the Differentiable Viewpoint. Princeton University Press, Princeton (1997) zbMATHGoogle Scholar
  12. 12.
    Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1991) zbMATHGoogle Scholar
  13. 13.
    Pumplün, S., Walcher, S.: On the zeros of polynomials over quaternions. Commun. Algebra 30, 4007–4018 (2002) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Oeding, L., Robeva, E., Sturmfels, B.: Decomposing tensors into frames. Adv. Appl. Math. 73, 125–153 (2016) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Qi, L.: Eigenvalues of a supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Qi, L.: Eigenvalues and invariants of tensors. J. Math. Anal. Appl. 325, 1363–1377 (2007) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Qi, L.: Transposes, L-eigenvalues and invariants of third order tensors (2017). arXiv:1704.01327. Preprint
  18. 18.
    Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018) zbMATHGoogle Scholar
  19. 19.
    Röhrl, H.: A theorem on nonassociative algebras and its application to differential equations. Manuscr. Math. 21, 181–187 (1977) zbMATHGoogle Scholar
  20. 20.
    Röhrl, H.: On the zeros of polynomials over arbitrary finite dimensional algebras. Manuscr. Math. 25, 359–390 (1978) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Röhrl, H., Walcher, S.: Projections of polynomial vector fields and the Poincaré sphere. J. Differ. Equ. 139, 22–40 (1997) zbMATHGoogle Scholar
  22. 22.
    Shafarevich, I.R.: Basic Algebraic Geometry. Springer, Berlin (1977) zbMATHGoogle Scholar
  23. 23.
    Virga, E.G.: Octupolar order in two dimensions. Eur. Phys. J. E 38, 63 (2015) Google Scholar
  24. 24.
    Walcher, S.: Algebras and Differential Equations. Hadronic Press, Palm Harbor (1991) zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Lehrstuhl A für MathematikRWTH AachenAachenGermany

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