Note on the dimension of certain algebraic sets of matrices
Jairo Bochi, Nicolas Gourmelon

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 is a nonempty algebraically closed subset of the affine space of complex matrices. Suppose that is column-invariant (i.e., belongingness to depends only on the column space of the matrix). Suppose is a vector subspace of that is not contained in the column space of any matrix in . Then . The proof is simple and relies on intersection theory of the grassmannians ("Schubert calculus").
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Taxonomy
TopicsAdvanced Topics in Algebra · Matrix Theory and Algorithms · Advanced Mathematical Theories and Applications
