### On the rank of 3x3x3 -tensors

(2013) LINEAR & MULTILINEAR ALGEBRA. 61(5). p.646-652- 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.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-2152936

- author
- Michel Lavrauw, Andrea Pavan and Corrado Zanella
- organization
- year
- 2013
- type
- journalArticle (original)
- publication status
- published
- subject
- keyword
- 15A72, ranks of tensors, 05E15, 14L24, 12K10
- journal title
- LINEAR & MULTILINEAR ALGEBRA
- Linear Multilinear Algebra
- volume
- 61
- issue
- 5
- pages
- 646 - 652
- Web of Science type
- Article
- Web of Science id
- 000313778400009
- JCR category
- MATHEMATICS
- JCR impact factor
- 0.7 (2013)
- JCR rank
- 95/302 (2013)
- JCR quartile
- 2 (2013)
- ISSN
- 0308-1087
- DOI
- 10.1080/03081087.2012.701299
- language
- English
- UGent publication?
- yes
- classification
- A1
- copyright statement
*I have transferred the copyright for this publication to the publisher*- id
- 2152936
- handle
- http://hdl.handle.net/1854/LU-2152936
- date created
- 2012-06-14 10:27:05
- date last changed
- 2016-12-19 15:44:36

@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}, }

- 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.