A simple estimation of the maximal rank of tensors with two slices by row and column operations, symmetrization and induction

TL;DR

This paper presents a straightforward method using row and column operations, symmetrization, and induction to estimate the maximal rank of 2 x n x n tensors, providing a tight bound that improves understanding in tensor theory and applications.

Contribution

It introduces a simple, elementary approach to determine the maximal rank of 2 x n x n tensors, simplifying previous eigenvalue-based methods.

Findings

Established a tight bound for the maximal rank of 2 x n x n tensors.

Demonstrated the effectiveness of elementary operations in tensor rank estimation.

Provided a new perspective that bypasses eigenvalue theories.

Abstract

The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a tensor model. Based on this classical theorem by Bosch we will show the tight bound for 2 x n x n tensors by simple row and column operations, symmetrization and mathematical induction, which has been given by several authors based on eigenvalue theories.