Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Decomposable tensors as a quadratic variety


Author: Robert Grone
Journal: Proc. Amer. Math. Soc. 64 (1977), 227-230
MSC: Primary 14M15; Secondary 15A69
MathSciNet review: 0472853
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Abstract: Let $ {V_i}$ be a finite dimensional vector space over a field F for each $ i = 1,2, \ldots ,m$, and let z be a tensor in $ {V_1} \otimes \cdots \otimes {V_m}$. In this paper a set of homogeneous quadratic polynomials in the coordinates of z is exhibited for which the associated variety is the set of decomposable tensors. In addition, a question concerning the maximal tensor rank in such a situation is answered, and an application to other symmetry classes of tensors is cited.


References [Enhancements On Off] (What's this?)

  • [1] Marvin Marcus, Finite dimensional multilinear algebra. Part II, Marcel Dekker, Inc., New York, 1975. Pure and Applied Mathematics, Vol. 23. MR 0401796
  • [2] Marvin Marcus, A dimension inequality for multilinear functions, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 217–224. MR 0332849
  • [3] William Watkins, Linear maps and tensor rank, J. Algebra 38 (1976), no. 1, 75–84. MR 0424855

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0472853-X
Keywords: Decomposable tensor, quadratic variety, quadratic Plücker relation, tensor rank
Article copyright: © Copyright 1977 American Mathematical Society