Euclidean Distance degrees of real algebraic groups

TL;DR

This paper investigates the Euclidean distance degree of real algebraic matrix groups, providing new formulas and recovering classical results, with a focus on the special linear groups.

Contribution

It introduces a new formula for the Euclidean distance degree of special linear groups and revisits classical results from an algebraic geometric perspective.

Findings

Derived a formula for the Euclidean distance degree of special linear groups

Revisited classical results on Euclidean distance degrees of algebraic groups

Provided an algebro-geometric approach to matrix approximation problems

Abstract

We study the problem of finding, in a real algebraic matrix group, the matrix closest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results; and among the new results that we prove is a formula for the Euclidean distance degree of special linear groups.