Perturbation theory for the spectral decomposition of Hermitian matrices

2018/09/24 by Marcus Carlsson, Carlsson, Marcus
Computer Science · Mathematics · Physics and Astronomy · #15A18 #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1809.09480

openalex publication_date 2018/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A and E be Hermitian self-adjoint matrices, where A is fixed and E a small perturbation. We study how the eigenvalues and eigenvectors of A+E depend on E, with the aim of obtaining first order formulas (and when possible also second order) that are explicitly computable in terms of the spectral decomposition of A and the entries in E. In particular we provide explicit Frechet type differentiability results. The findings can be seen as an extension of the Rayleigh-Schrödinger coefficients for analytic expansions of one-dimensional perturbations.

Citations

• The Rotation of Eigenvectors by a Perturbation. III

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