  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.
