Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eigenvalue stability of Hermitian and normal matrices

Adam Parusiński, Armin Rainer

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.

Eigenvalue stability of Hermitian and normal matrices

Abstract

The ordered eigenvalues define a Lipschitz map on the real vector space of Hermitian matrices. We prove that this map acts continuously, but not uniformly continuously, by superposition on the Sobolev spaces , for all , on bounded open domains. For , 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.
Paper Structure (58 sections, 68 theorems, 330 equations)

This paper contains 58 sections, 68 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. Hermitian matrices
  3. Normal matrices
  4. Optimality of the results
  5. Singular values
  6. Applications
  7. Functions defined on the spectrum or the singular values
  8. Condition numbers
  9. Surface area
  10. Compact self-adjoint operators
  11. Structure of the paper
  12. Notation
  13. Function spaces
  14. Lebesgue spaces
  15. Sobolev spaces
  16. ...and 43 more sections

Key Result

Theorem 1.1

[t]thm:eigW Let $1 \le q < \infty$. Let $U \subseteq \mathbb{R}^m$ be open and bounded. Then the characteristic map is continuous.

Theorems & Definitions (122)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • ...and 112 more