On non-commutative rank and tensor rank

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.