On non-commutative rank and tensor rank

TL;DR

This paper explores the relationship between commutative and non-commutative tensor ranks, providing examples and equations that establish bounds and properties relevant to tensor border rank and its algebraic characterization.

Contribution

It introduces new bounds and equations for tensor border rank using non-commutative rank techniques and concavity of tensor blow-ups, extending previous results and providing alternative proofs.

Findings

The ratio of non-commutative to commutative rank can approach 2.

New equations for tensors of border rank 2m-4 in odd m cases.

Alternative proof of the regularity lemma by Ivanyos et al.

Abstract

We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give nontrivial equations for the tensors of border rank at most $2m-3$ in $K^m\otimes K^m\otimes K^m$ if $m$ is even. He also gave such equations for tensors of border rank at most $2m-5$ in $K^m\otimes K^m\otimes K^m$ if $m$ is odd. Using concavity of tensor blow-ups we show non-trivial equations for tensors of border rank $2m-4$ in $K^m \otimes K^m \otimes K^m$ for odd $m$ for any field $K$ of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam.