 Author
 Michel Lavrauw (UGent) , Andrea Pavan and Corrado Zanella
 Organization
 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: http://hdl.handle.net/1854/LU2152936
 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 = {03081087}, journal = {LINEAR \& MULTILINEAR ALGEBRA}, keyword = {15A72,ranks of tensors,05E15,14L24,12K10}, language = {eng}, number = {5}, pages = {646652}, 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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