The Euclidean Distance Degree of Orthogonally Invariant Matrix Varieties

TL;DR

This paper demonstrates that the Euclidean distance degree of orthogonally invariant matrix varieties can be computed by examining their diagonal restrictions, simplifying calculations in practical scenarios.

Contribution

It establishes a key equivalence between the Euclidean distance degree of matrix varieties and their diagonal restrictions, aiding computational efficiency.

Findings

Euclidean distance degree equals that of diagonal restriction

Simplifies calculations for orthogonally invariant varieties

Applicable to practical matrix analysis scenarios

Abstract

We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.