Note on the dimension of certain algebraic sets of matrices

TL;DR

This paper proves a lemma about the dimension of algebraic sets of matrices, specifically relating to column-invariance and subspace containment, with applications in algebraic geometry and matrix theory.

Contribution

It introduces a new dimension bound for algebraic sets of matrices based on column-invariance and subspace conditions, using intersection theory.

Findings

Provides a lower bound on codimension of algebraic sets of matrices

Uses Schubert calculus for proof

Applicable in algebraic geometry and matrix analysis

Abstract

In this short note we prove a lemma about the dimension of certain algebraic sets of matrices. This result is needed in our paper arXiv:1201.1672. The result presented here has also applications in other situations and so it should appear as part of a larger work. The statement of the lemma goes as follows: Suppose $X$ is a nonempty algebraically closed subset of the affine space of $n \times m$ complex matrices. Suppose that $X$ is column-invariant (i.e., belongingness to $X$ depends only on the column space of the matrix). Suppose $E$ is a vector subspace of $\mathbb{C}^n$ that is not contained in the column space of any matrix in $X$. Then $codim(X) \geq m + 1 - dim(E)$. The proof is simple and relies on intersection theory of the grassmannians ("Schubert calculus").