Matrix tuples with linearly dependent invariant subspaces

TL;DR

This paper characterizes the algebraic hypersurface of matrix tuples with invariant subspaces that do not span the entire space, providing the explicit equation and generalizing prior results.

Contribution

It derives the explicit equation of the hypersurface of matrix tuples with certain invariant subspace properties, extending previous work.

Findings

Identified the algebraic hypersurface of such matrix tuples.

Derived the explicit polynomial equation defining this hypersurface.

Generalized earlier results to a broader class of matrix tuples.

Abstract

The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes previous joint work.