To what extent is a large space of matrices not closed under the product?

TL;DR

This paper investigates the algebraic structure of large subspaces of matrices over a field, demonstrating that under certain conditions, these subspaces can generate the entire matrix algebra through products, with specific results for hyperplanes.

Contribution

The authors strengthen classical results by showing that matrix subspaces of certain codimensions can generate the full algebra via products, including decompositions into products of elements in the subspace.

Findings

M_n(K) is spanned by products of matrices in V

Every matrix in M_n(K) can be decomposed into products of matrices in V

For hyperplanes V in M_n(K), every matrix is a product of two elements in V

Abstract

Let K denote a field. Given an arbitrary linear subspace V of M_n(K) of codimension lesser than n-1, a classical result states that V generates the K-algebra M_n(K). Here, we strengthen this in three ways: we show that M_n(K) is spanned by the products of the form AB with A and B in V; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear hyperplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.