Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
skip to main content
article

Eigenvectors of Tensors--A Primer

Published: 01 August 2019 Publication History

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.

References

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

Recommendations

Comments

Information & Contributors

Information

Published In

cover image Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications  Volume 162, Issue 1
August 2019
179 pages

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 August 2019

Author Tags

  1. 14Q05
  2. 15A69
  3. 34A05
  4. 76A15
  5. Bezout theorem
  6. Brouwer degree
  7. Liquid crystal

Qualifiers

  • Article

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 0
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 28 Jan 2025

Other Metrics

Citations

View Options

View options

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media