Hermitian Distance Degree of Unitary-Invariant Matrix Varieties

TL;DR

This paper investigates the Hermitian distance degree of unitary-invariant matrix varieties, linking it to the Euclidean distance degree of their singular value sets, and introduces a Hermitian slicing theorem for critical point analysis.

Contribution

It establishes a connection between Hermitian and Euclidean distance degrees for invariant matrix varieties and introduces a Hermitian slicing theorem for critical point computation.

Findings

Hermitian distance degree equals the Euclidean distance degree of the singular value variety.

Critical points on the matrix variety can be obtained from singular value slices.

A Hermitian analogue of the Eckart-Young theorem is recovered.

Abstract

We study the Hermitian distance degree, a real enumerative invariant counting critical points of the squared Hermitian distance function, for matrix varieties invariant under left and right unitary actions. For such a variety \(M \subset \mathbb{C}^{n\times t}\), we prove that its Hermitian distance degree equals the real Euclidean distance degree of the associated absolutely symmetric variety of singular values. Equivalently, for a generic data matrix, Hermitian distance critical points on \(M\) are obtained by lifting Euclidean distance critical points from the singular-value slice. We also establish a Hermitian slicing theorem, paralleling the Bik--Draisma principle, which reduces the critical point count to a diagonal slice. As a motivating example, we recover a geometric Hermitian analogue of the Eckart-Young theorem.