The classification of large spaces of matrices with bounded rank

Abstract

Given an arbitrary (commutative) field $\mathbb{K}$, let V be a linear subspace of $M_n (\mathbb{K})$ consisting of matrices of rank less than or equal to some r∈[[1, n−1]]. A theorem of Atkinson and Lloyd states that, if dimV > nr − r + 1 and $\# \mathbb{K} > r$, then either all the matrices of V vanish everywhere on some common (n − r)-dimensional subspace of $\mathbb{K}^n$, or it is true of the matrices of the transposed space V ^T. Using a new approach, we prove that the restriction on the cardinality of the underlying field is unnecessary. We also show that the results of Atkinson and Lloyd in the case dimV = nr−r+1 hold for any field, except in the special case when n = 3, r = 2 and K≃F ². In that exceptional situation, we classify all the exceptional spaces up to equivalence. Similar theorems of Beasley for rectangular matrices are also extended to all fields. Finally, we extend Atkinson, Lloyd and Beasley’s classification theorems to a range of high dimensions which almost doubles theirs, under the assumption that $\# \mathbb{K} > r$.