Eigenvalue perturbation

In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues 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.

The derivations in this article are essentially self-contained and can be found in many texts on numerical linear algebra or numerical functional analysis.

Suppose we have solutions to the generalized eigenvalue problem,

where ${\displaystyle \mathbf {K} _{0}}$ and ${\displaystyle \mathbf {M} _{0}}$ are matrices. That is, we know the eigenvalues λ_0i and eigenvectors x_0i for i = 1, ..., N. Now suppose we want to change the matrices by a small amount. That is, we want to find the eigenvalues and eigenvectors of

...
Wikipedia