On the Continuity of Schur-Horn Mapping
Hengzhun Chen, Yingzhou Li
TL;DR
The paper studies continuity of the Schur-Horn mapping for symmetric and Hermitian matrices with fixed diagonals by developing a perturbation framework anchored in majorization. It introduces strong Schur-Horn continuity and proves SH continuity for diagonal, symmetric, and Hermitian matrices via constructive, near-identity orthogonal/unitary transformations and block-wise majorization arguments. The results enable explicit, distance-controlled adjustments of eigenvalues to match diagonals, and they are applied to oblique-manifold optimization problems in quantum computing (e.g., qOMM) to ensure feasible perturbations along descent paths. This continuity theory provides a rigorous foundation for landscape analysis and perturbation-based optimization on constrained matrix manifolds with potential broader impact in quantum information and convex optimization.
Abstract
The Schur-Horn theorem is a well-known result that characterizes the relationship between the diagonal elements and eigenvalues of a symmetric (Hermitian) matrix. In this paper, we extend this theorem by exploring the eigenvalue perturbation of a symmetric (Hermitian) matrix with fixed diagonals, which is referred to as the continuity of the Schur-Horn mapping. We introduce a concept called strong Schur-Horn continuity, characterized by minimal constraints on the perturbation. We demonstrate that several categories of matrices exhibit strong Schur-Horn continuity. Leveraging this notion, along with a majorization constraint on the perturbation, we prove the Schur-Horn continuity for general symmetric (Hermitian) matrices. The Schur-Horn continuity finds applications in oblique manifold optimization related to quantum computing.