On the Continuity of Schur-Horn Mapping

• URL: http://arxiv.org/abs/2407.00701v1
• Date: Sun, 30 Jun 2024 13:48:02 GMT
• Title: On the Continuity of Schur-Horn Mapping
• Abstract summary: We introduce a concept called strong Schur-Horn continuity, characterized by minimal constraints on the perturbation. We prove the Schur-Horn continuity for general symmetric (Hermitian) matrices. The continuity finds applications in oblique manifold optimization related to quantum computing.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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