Minimum (maximum) rank of tensors and the sign nonsingular tensors

TL;DR

This paper introduces new concepts of minimum and maximum tensor ranks, explores their properties, and characterizes sign nonsingular tensors, providing theoretical insights into tensor invertibility and rank relations.

Contribution

It defines and analyzes minimum and maximum tensor ranks, establishes conditions for sign nonsingular tensors, and characterizes tensors with sign inverses, advancing tensor theory.

Findings

Minimum tensor rank equals 1 under certain conditions

Maximum tensor rank is at least the term rank

Sign nonsingular tensors have minimum rank at least their dimension

Abstract

In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not less than the term rank. We also prove that the minimum rank of a sign nonsingular tensor is not less than the dimension of it. And we get some characterizations of a tensor having sign left or sign right inverses.