Bauer-Fike theorem
The Bauer-Fike theorem measures how much the eigenvalues of a diagonalizable matrix can move when you nudge the matrix. The surprise it captures is that the answer depends not just on the size of the nudge, but on how skewed the eigenvectors are. Nearly parallel eigenvectors make eigenvalues hypersensitive.
Suppose A = V D V^-1 is diagonalizable, and you perturb it to A + E. The theorem says every eigenvalue mu of A + E lies within distance kappa(V) ||E|| of some eigenvalue of A, where kappa(V) = ||V|| ||V^-1|| is the condition number of the eigenvector matrix. In symbols, min over lambda of |mu - lambda| <= kappa(V) ||E||, using a consistent norm such as the 2-norm.
The factor kappa(V) is the eigenvalue conditioning. If A is normal — symmetric, Hermitian, or unitary — its eigenvectors can be chosen orthonormal, so V is unitary, kappa(V) = 1, and eigenvalues move at most by ||E||: as good as it gets. But for a strongly non-normal matrix kappa(V) can be enormous, and a tiny perturbation can throw eigenvalues far.
This explains a notorious numerical fact: defective or near-defective matrices have wildly unstable eigenvalues, and computing them accurately is hard not because of bad algorithms but because the problem itself is ill-conditioned. Bauer-Fike is the quantitative form of that warning, and the pseudospectrum is its picture.
The whole story in one line: perturbation size times eigenvector conditioning bounds the eigenvalue shift.
For symmetric or Hermitian matrices kappa(V) = 1, so Bauer-Fike collapses into Weyl's theorem: eigenvalues move at most by the norm of the perturbation. Symmetry buys you perfectly conditioned eigenvalues.