On the Continuity of Schur-Horn Mapping
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.
Keywords
Cite
@article{arxiv.2407.00701,
title = {On the Continuity of Schur-Horn Mapping},
author = {Hengzhun Chen and Yingzhou Li},
journal= {arXiv preprint arXiv:2407.00701},
year = {2026}
}