Updating eigenvectors for small perturbation of source matrix

Question

I have eigenvectors for matrix A, and I'm interested in eigenvectors of A+eps, where eps is a matrix of small norm. Is there an efficient procedure for this problem?

Answer 1

This is known as eigenvalue perturbation problem.

For the special case of symmetric positive definite matrices $A$ and $\epsilon$ with distinct eigenvalues, an approximation is as follows:

Let $u_1, u_2, \ldots, u_n$ be eigenvectors of $A$ such that $u_i^TAu_j = \delta_{ij}$ with corresponding eigenvalues $\lambda_1, \ldots \lambda_n$.

That is $$Au_i = \lambda_i u_i$$

The new set of eigenvectors, $v_1, \ldots, v_n$ can be expressed as

$$v_i=u_i + \sum_{j=1\\ j\neq i}^{n} \frac{u_j^T\epsilon u_i}{\lambda_i - \lambda_j}u_j$$

Wikipedia references: eigenvalue perturbation.

Remark: I notice that the wikipedia page does not address what happens when $\lambda_i = \lambda_j$.