Eigenvalue perturbation

In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system ${\displaystyle Ax=\lambda x}$ that is perturbed from one with known eigenvectors and eigenvalues ${\displaystyle A_{0}x_{0}=\lambda _{0}x_{0}}$. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues ${\displaystyle x_{0i},\lambda _{0i},i=1,\dots n}$ are to changes in the system. This type of analysis was popularized by Lord Rayleigh, in his investigation of harmonic vibrations of a string perturbed by small inhomogeneities.^[1]

The derivations in this article are essentially self-contained and can be found in many texts on numerical linear algebra or numerical functional analysis. This article is focused on the case of the perturbation of a simple eigenvalue (see in multiplicity of eigenvalues).