On the perturbation series for eigenvalues and eigenprojections

2019/10/18 by Martin Wahl, Wahl, Martin
Computer Science · Mathematics · #15A42 #47A55 #47H40 #62H25 #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Numerical methods in inverse problems #Probability (math.PR) #Spectral Theory in Mathematical Physics #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1910.08460

openalex publication_date 2019/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A standard perturbation result states that perturbed eigenvalues and eigenprojections admit a perturbation series provided that the operator norm of the perturbation is smaller than a constant times the corresponding eigenvalue isolation distance. In this paper, we show that the same holds true under a weighted condition, where the perturbation is symmetrically normalized by the square-root of the reduced resolvent. This weighted condition originates in random perturbations where it leads to significant improvements.

Citations

• Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

Related

• First-order Perturbation Theory for Eigenvalues and Eigenvectors
• Coefficients of almost-degenerate density matrix perturbation theory for eigenvalue problems
• Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matrices II
• Notes on Rank One Perturbed Resolvent. Perturbation of Isolated\n Eigenvalue
• Rank 1 perturbations in random matrix theory -- a review of exact results