×

Characteristic polynomial and higher order traces of third order three dimensional tensors. (English) Zbl 1416.15007

Summary: Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
53A45 Differential geometric aspects in vector and tensor analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen H, Qi L, Song Y. Column sufficient tensors and tensor complementarity problems. Front Math China, 2018, 13: 255-276 · Zbl 1418.90253 · doi:10.1007/s11464-018-0681-4
[2] Chen H, Wang Y. On computing minimal H-eigenvalue of sign-structured tensors. Front Math China, 2017, 12: 1289-1302 · Zbl 1401.65037 · doi:10.1007/s11464-017-0645-0
[3] Cox D, Little J, O’Shea D. Using Algebraic Geometry. New York: Springer-Verlag, 1998 · Zbl 0920.13026 · doi:10.1007/978-1-4757-6911-1
[4] Horn R A, Johnson C R. Matrix Analysis. New York: Cambridge Univ Press, 1985 · Zbl 0576.15001 · doi:10.1017/CBO9780511810817
[5] Hu, S., Spectral symmetry of uniform hypergraphs (2014)
[6] Hu, S., Symmetry of eigenvalues of Sylvester matrices and tensors (2019)
[7] Hu S, Huang Z, Ling C, Qi L. On determinants and eigenvalue theory of tensors. J Symbolic Comput, 2013, 50: 508-531 · Zbl 1259.15038 · doi:10.1016/j.jsc.2012.10.001
[8] Hu, S.; Lim, L-H, Spectral symmetry of uniform hypergraphs (2014)
[9] Hu S, Ye K. Multiplicities of tensor eigenvalues. Commun Math Sci, 2016, 14: 1049-1071 · Zbl 1353.15007 · doi:10.4310/CMS.2016.v14.n4.a9
[10] Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302-1324 · Zbl 1125.15014 · doi:10.1016/j.jsc.2005.05.007
[11] Shafarevich I R. Basic Algebraic Geometry. Berlin: Springer-Verlag, 1977 · Zbl 0362.14001
[12] Shao J-Y, Qi L, Hu S. Some new trace formulas of tensors with applications in spectral hypergraph theory. Linear Multilinear Algebra, 2015, 63: 871-992 · Zbl 1310.15042
[13] Sturmfels B. Solving Systems of Polynomial Equations. CBMS Reg Conf Ser Math, No 97. Providence: Amer Math Soc, 2002 · Zbl 1101.13040 · doi:10.1090/cbms/097
[14] Wang X, Wei Y. H-tensors and nonsingular H-tensors. Front Math China, 2016, 11: 557-575 · Zbl 1381.15018 · doi:10.1007/s11464-015-0495-6
[15] Wang Y, Zhang K, Sun H. Criteria for strong H-tensors. Front Math China, 2016, 11: 577-592 · Zbl 1381.15019 · doi:10.1007/s11464-016-0525-z
[16] Yang Q, Zhang L, Zhang T, Zhou G. Spectral theory of nonnegative tensors. Front Math China, 2013, 8: 1 · Zbl 1261.15002 · doi:10.1007/s11464-012-0273-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.