Typical rank of $m\times n\times (m-1)n$ tensors with $3\le m\le n$ over the real number field

Abstract

Tensor type data are used recently in various application fields, and then a typical rank is important. Let $3\le m\le n$. We study typical ranks of $m\times n\times (m-1)n$ tensors over the real number field. Let $\rho$ be the Hurwitz-Radon function defined as $\rho (n)=2^b+8c$ for nonnegative integers $a,b,c$ such that $n=(2a+1)2^{b+4c}$ and $0\le b<4$. If $m\le \rho (n)$, then the set of $m\times n\times (m-1)n$ tensors has two typical ranks $(m-1)n,(m-1)n+1$. In this paper, we show that the converse is also true: if $m>\rho (n)$, then the set of $m\times n\times (m-1)n$ tensors has only one typical rank $(m-1)n$.

20 pages

References in corpus (1)

• Rank of 3-tensors with 2 slices and Kronecker canonical forms

Cited by in corpus (1)

• Upper bound of typical ranks of m x n x ((m-1)n-1) tensors over the real number field