### Abstract

In various application fields, tensor type data are used recently and then a typical rank is important. There may be more than one typical ranks over the real number field. It is well known that the set of 2×n×n tensors has two typical ranks n,n+1 for n≥2, that the set of 3×4×8 tensors has two typical ranks 8,9, and that the set of 4×4×12 tensors has two typical ranks 12,13. In this paper, we show that the set of m×n×(m-1)n tensors with m≤n has two typical ranks (m-1)n,(m-1)n+1 if m≤ρ(n), where ρ is the Hurwitz-Radon function defined as ρ(n)=2^{b}+8c for nonnegative integers a,b,c such that n=(2a+1)2^{b+4c} and 0≤b<4.

Original language | English |
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Pages (from-to) | 953-958 |

Number of pages | 6 |

Journal | Linear Algebra and Its Applications |

Volume | 438 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 15 2013 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

Sumi, T., Sakata, T., & Miyazaki, M. (2013). Typical ranks for m × n × (m - 1) n tensors with m ≤ n.

*Linear Algebra and Its Applications*,*438*(2), 953-958. https://doi.org/10.1016/j.laa.2011.08.009