Distance Optimization in the Grassmannian of Lines

TL;DR

This paper develops a new metric algebraic geometry framework for the Grassmannian of lines, introducing the Grassmann distance degree as an invariant and analyzing it for Schubert varieties and other models.

Contribution

It introduces the Grassmann distance degree as a novel invariant and explores its properties for various subvarieties of the Grassmannian of lines.

Findings

GD degree computed for Schubert varieties

New invariant links algebraic geometry and linear algebra

Framework applicable to line varieties in projective space

Abstract

The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Pl\"ucker coordinates to projection coordinates. We develop metric algebraic geometry for varieties of lines in this linear algebra setting. The Grassmann distance (GD) degree is introduced as a new invariant for subvarieties of a Grassmannian. We study the GD degree for Schubert varieties and other models.