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.

翻译：Schur-Horn定理是一个著名结论，它刻画了对称（Hermitian）矩阵的对角元与特征值之间的关系。本文通过探究固定对角元的对称（Hermitian）矩阵的特征值扰动，扩展了该定理，这被称为Schur-Horn映射的连续性。我们引入了一个称为强Schur-Horn连续性的概念，其特点是对扰动施加了最小约束。我们证明了几类矩阵具有强Schur-Horn连续性。利用这一概念，并结合扰动上的优超约束，我们证明了对于一般对称（Hermitian）矩阵的Schur-Horn连续性。Schur-Horn连续性在与量子计算相关的斜流形优化中具有应用。