Perfect type of n-tensors

TL;DR

This paper characterizes the perfect types of n-tensors, which are tensor types with a typical rank equal to the maximum dimension among their modes, over the complex numbers.

Contribution

It provides a complete determination of which tensor types are perfect, advancing understanding of tensor ranks in multilinear algebra.

Findings

Identifies conditions for perfect tensor types

Determines the set of perfect n-tensor types

Advances tensor rank theory in multilinear algebra

Abstract

In various application fields, tensor type data are used recently and then a typical rank is important. Although there may be more than one typical ranks over the real number field, a generic rank over the complex number field is the minimum number of them. The set of $n$-tensors of type $p_1\times p_2\times\cdots\times p_n$ is called perfect, if it has a typical rank $\max(p_1,\ldots,p_n)$. In this paper, we determine perfect types of $n$-tensor.