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On the rank of 3x3x3 -tensors

(2013) LINEAR & MULTILINEAR ALGEBRA. 61(5). p.646-652
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Abstract
Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor t in U???V???W is the minimum dimension of a subspace of U???V???W containing t and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.
Keywords
15A72, ranks of tensors, 05E15, 14L24, 12K10

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Citation

Please use this url to cite or link to this publication:

Chicago
Lavrauw, Michel, Andrea Pavan, and Corrado Zanella. 2013. “On the Rank of 3x3x3 -tensors.” Linear & Multilinear Algebra 61 (5): 646–652.
APA
Lavrauw, M., Pavan, A., & Zanella, C. (2013). On the rank of 3x3x3 -tensors. LINEAR & MULTILINEAR ALGEBRA, 61(5), 646–652.
Vancouver
1.
Lavrauw M, Pavan A, Zanella C. On the rank of 3x3x3 -tensors. LINEAR & MULTILINEAR ALGEBRA. 2013;61(5):646–52.
MLA
Lavrauw, Michel, Andrea Pavan, and Corrado Zanella. “On the Rank of 3x3x3 -tensors.” LINEAR & MULTILINEAR ALGEBRA 61.5 (2013): 646–652. Print.
@article{2152936,
  abstract     = {Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor t in U???V???W is the minimum dimension of a subspace of U???V???W containing t and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.},
  author       = {Lavrauw, Michel and Pavan, Andrea and Zanella, Corrado },
  issn         = {0308-1087},
  journal      = {LINEAR \& MULTILINEAR ALGEBRA},
  keyword      = {15A72,ranks of tensors,05E15,14L24,12K10},
  language     = {eng},
  number       = {5},
  pages        = {646--652},
  title        = {On the rank of 3x3x3 -tensors},
  url          = {http://dx.doi.org/10.1080/03081087.2012.701299},
  volume       = {61},
  year         = {2013},
}

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