Large spaces of symmetric or alternating matrices with bounded rank

Abstract

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to $r$, and we have classified the spaces having that maximal dimension. In this article, provided that $\mathbb{K}$ has more than two elements, we extend this classification to spaces whose dimension is close to the maximal one: this generalizes a result of Loewy. We also prove a similar result on spaces of alternating matrices with bounded rank, with no restriction on the cardinality of the underlying field.

41 pages (version 2, minor errors corrected)

References in corpus (3)

• The affine preservers of non-singular matrices
• Large spaces of bounded rank matrices revisited
• Range-compatible homomorphisms on spaces of symmetric or alternating matrices