Eigenvalue stability of Hermitian and normal matrices

TL;DR

This paper investigates the stability and continuity properties of eigenvalues of Hermitian and normal matrices as functions in Sobolev spaces, revealing nuanced behaviors depending on the function space and extending results to infinite-dimensional operators.

Contribution

It establishes the Lipschitz continuity of eigenvalue maps in Sobolev spaces for Hermitian matrices and extends these stability results to normal matrices and operators in Hilbert space.

Findings

Eigenvalue maps are Lipschitz continuous on Sobolev spaces for Hermitian matrices.

Continuity of eigenvalue maps fails at the supremum Sobolev space ($q= finite$).

Applications include stability of singular values, condition numbers, and eigenvalue graph surface areas.

Abstract

The ordered eigenvalues define a Lipschitz map on the real vector space of Hermitian $d \times d$ matrices. We prove that this map acts continuously, but not uniformly continuously, by superposition on the Sobolev spaces $W^{1,q}$, for all $1 \le q < \infty$, on bounded open domains. For $q=\infty$, the action is still well-defined and bounded but not continuous. We show that this stability result extends to normal matrices, where the eigenvalues are naturally interpreted as multivalued Sobolev functions in the sense of Almgren. Several applications are given, including the stability of singular values, condition numbers of matrices, surface area of eigenvalue graphs, and compact self-adjoint operators in Hilbert space.