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I was asked recently (in relation to my recent work with Van Vu on the spectral theory of random matrices) to explain some standard heuristics regarding how the eigenvalues of an matrix A behave under small perturbations. These heuristics can be summarised as follows:
- For normal matrices (and in particular, unitary or self-adjoint matrices), eigenvalues are very stable under small perturbations. For more general matrices, eigenvalues can become unstable if there is pseudospectrum present.
- For self-adjoint (Hermitian) matrices, eigenvalues that are too close together tend to repel quickly from each other under such perturbations. For more general matrices, eigenvalues can either be repelled or be attracted to each other, depending on the type of perturbation.
In this post I would like to briefly explain why these heuristics are plausible.