### Abstract

Let m, n and p be integers with (Formula presented.) and (Formula presented.). We showed in previous papers that if (Formula presented.), then typical ranks of (Formula presented.) -tensors over the real number field are p and p+1 if and only if there exists a non-singular bilinear map (Formula presented.). We also showed that the ‘if’ part is also valid in the case where (Formula presented.). In this paper, we consider the case where (Formula presented.) and show that the typical ranks of (Formula presented.) -tensors over the real number field are p and p+1 in several cases including the case where there is no non-singular bilinear map (Formula presented.). In particular, we show that the ‘only if’ part of the above mentioned fact is not valid for the case (Formula presented.).

Original language | English |
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Journal | Linear and Multilinear Algebra |

DOIs | |

Publication status | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Linear and Multilinear Algebra*. https://doi.org/10.1080/03081087.2019.1637811

**Typical ranks of semi-tall real 3-tensors.** / Sumi, Toshio; Miyazaki, Mitsuhiro; Sakata, Toshio.

Research output: Contribution to journal › Article

*Linear and Multilinear Algebra*. https://doi.org/10.1080/03081087.2019.1637811

}

TY - JOUR

T1 - Typical ranks of semi-tall real 3-tensors

AU - Sumi, Toshio

AU - Miyazaki, Mitsuhiro

AU - Sakata, Toshio

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let m, n and p be integers with (Formula presented.) and (Formula presented.). We showed in previous papers that if (Formula presented.), then typical ranks of (Formula presented.) -tensors over the real number field are p and p+1 if and only if there exists a non-singular bilinear map (Formula presented.). We also showed that the ‘if’ part is also valid in the case where (Formula presented.). In this paper, we consider the case where (Formula presented.) and show that the typical ranks of (Formula presented.) -tensors over the real number field are p and p+1 in several cases including the case where there is no non-singular bilinear map (Formula presented.). In particular, we show that the ‘only if’ part of the above mentioned fact is not valid for the case (Formula presented.).

AB - Let m, n and p be integers with (Formula presented.) and (Formula presented.). We showed in previous papers that if (Formula presented.), then typical ranks of (Formula presented.) -tensors over the real number field are p and p+1 if and only if there exists a non-singular bilinear map (Formula presented.). We also showed that the ‘if’ part is also valid in the case where (Formula presented.). In this paper, we consider the case where (Formula presented.) and show that the typical ranks of (Formula presented.) -tensors over the real number field are p and p+1 in several cases including the case where there is no non-singular bilinear map (Formula presented.). In particular, we show that the ‘only if’ part of the above mentioned fact is not valid for the case (Formula presented.).

UR - http://www.scopus.com/inward/record.url?scp=85068687563&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068687563&partnerID=8YFLogxK

U2 - 10.1080/03081087.2019.1637811

DO - 10.1080/03081087.2019.1637811

M3 - Article

AN - SCOPUS:85068687563

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

ER -