TY - JOUR

T1 - About the maximal rank of 3-tensors over the real and the complex number field

AU - Sumi, Toshio

AU - Miyazaki, Mitsuhiro

AU - Sakata, Toshio

N1 - Funding Information:
This work was supported in part by Grant-in-Aid for Scientific Research (B) (No. 20340021) of the Japan Society for the Promotion of Science.

PY - 2010/8

Y1 - 2010/8

N2 - Tensor data are becoming important recently in various application fields. In this paper, we consider themaximal rank problem of 3-tensors and extend Atkinson and Stephens' and Atkinson and Lloyd's results over the real number field. We also prove the assertion of Atkinson and Stephens: max.rankR(m, n, p) ≤ m + [p/2]n, max.rankR(n, n, p) ≤ (p+1)n/2 if p is even, max.rankF(n, n, 3) ≤ 2n-1 if F = C or n is odd, and max.rankF(m, n, 3) ≤ m +n-1 if m < n where F stands for ℝ or ℂ .

AB - Tensor data are becoming important recently in various application fields. In this paper, we consider themaximal rank problem of 3-tensors and extend Atkinson and Stephens' and Atkinson and Lloyd's results over the real number field. We also prove the assertion of Atkinson and Stephens: max.rankR(m, n, p) ≤ m + [p/2]n, max.rankR(n, n, p) ≤ (p+1)n/2 if p is even, max.rankF(n, n, 3) ≤ 2n-1 if F = C or n is odd, and max.rankF(m, n, 3) ≤ m +n-1 if m < n where F stands for ℝ or ℂ .

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U2 - 10.1007/s10463-010-0294-5

DO - 10.1007/s10463-010-0294-5

M3 - Article

AN - SCOPUS:77958070995

VL - 62

SP - 807

EP - 822

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 4

ER -