On the Continuity of Schur-Horn Mapping
<|reference_start|>On the Continuity of Schur-Horn Mapping: 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.<|reference_end|>
@article{chen2024on,
    title={On the Continuity of Schur-Horn Mapping},
    author={Hengzhun Chen, Yingzhou Li},
    journal={arXiv preprint arXiv:2407.00701},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.00701},
    primaryClass={math.NA cs.NA quant-ph}
}