  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.
