Continuity in finding eigenvectors

Question

I'm wondering whether there's "continuity" in the eigen vectors of different matrices corresponding to appropriate eigenvalues.

For instance, if we change certain elements in a matrix, can we measure how does that change the eigen vectors?(maybe we can measure difference in normalized eigen vectors by angle?)

Thank you!

Answer 1

Look up "perturbation theory". It's simplest in the case of hermitian matrices with distinct eigenvalues, but can be done more generally as well.

Answer 2

I would say no. For instance, the two-dimensional identity matrix has any vector for eigenvector, and eigenvalue 1. If we rotate it ever so slightly, it will have no (real) eigenvectors.