关于Schur-Horn映射的连续性
On the Continuity of Schur-Horn Mapping
AI总结：
本文研究了对称矩阵在固定对角线下的特征值扰动，提出强Schur-Horn连续性概念，并证明了其在量子计算优化中的应用。
摘要：
Schur-Horn定理是一个著名的结论，它刻画了对称（正交）矩阵的对角线元素与特征值之间的关系。在本文中，我们通过研究具有固定对角线的对称（正交）矩阵的特征值扰动，扩展了这一定理，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，其特点是对于扰动有最小的约束。我们证明了几类矩阵表现出强Schur-Horn连续性。利用这一概念，结合对扰动的主化约束，我们证明了一般对称（正交）矩阵的Schur-Horn连续性。Schur-Horn连续性在与量子计算相关的斜 manifold 优化中找到了应用。
英文摘要：
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.
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