Applications of perturbation expansion of eigenvectors

Question

Partially citing this thread, consider $A_{\varepsilon}=A+\varepsilon V$, a perturbation of $A$, where $\varepsilon$ is a small number (All matrices are symmetric).

A known result derived in some books that I've read is the following perturbation expansion for the eigenvectors of $A_{\varepsilon}$ ($e_j$ are the eigenvectors of $A$):

$$e_{j}' = e_{j} +\epsilon \sum_{k=1\;(k\ne j)}^{k} \frac{(Ve_j,e_k)}{\lambda_j-\lambda_k}e_k.$$

My question is are there any applications, theoretical but most importantly practical, where this expansion is used?

Answer 1

The analytic dependence of an eigenvalue from a perturbation of the linear operator is a question that has many applications in the study of dynamical systems, and it extension to linear operators in function (Hilbert) spaces is very important in quantum mechanics applications. A classical book on this subject is here.