On the Continuity of Schur-Horn Mapping

TL;DR

This paper extends the Schur-Horn theorem by analyzing eigenvalue perturbations with fixed diagonals, introducing strong continuity concepts, and applying these results to quantum computing optimization.

Contribution

It introduces the concept of strong Schur-Horn continuity and proves Schur-Horn continuity for general matrices under perturbation constraints.

Findings

Several matrix categories exhibit strong Schur-Horn continuity.

Schur-Horn continuity is established for general symmetric matrices.

Applications demonstrated in quantum computing optimization.

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.