Hermitian Distance Degree of Unitary-Invariant Matrix Varieties

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.

💡 Research Summary

The paper introduces and studies the Hermitian distance degree (HDdeg), a real enumerative invariant that counts the real critical points of the squared Hermitian distance function on matrix varieties that are invariant under left and right unitary actions. By identifying the complex matrix space (\mathbb C^{n\times t}) with the real vector space (\mathbb R^{2nt}) and using the real part of the Hermitian inner product as the Euclidean metric, the authors show that HDdeg is precisely the real Euclidean distance degree (R‑EDdeg) of the underlying real algebraic set.

A central structural observation is that any unitary‑invariant matrix variety (M) can be described completely by an absolutely symmetric set (S\subset\mathbb R^{n}) of singular‑value vectors: (S=\sigma(M)={x\mid\operatorname{diag}(x)\in M}) and (M=\sigma^{-1}(S)={U\operatorname{diag}(x)V^{*}\mid x\in S,;U\in U(n),V\in U(t)}). The set (S) is invariant under all signed permutations, which captures the full symmetry of (M).

The main theorem (Theorem 23) proves that for a generic data matrix (Y=U\operatorname{diag}(y)V^{*}) with distinct non‑zero singular values, the Hermitian distance degree of (M) at (Y) equals the real Euclidean distance degree of the singular‑value variety (S) at the vector (y):
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