Eigenvalues of a Hermitian matrix are real, ordered, and continuous functions of the matrix entries. But how smooth are they? As the matrix changes, do eigenvalues move smoothly or can they jump?
The paper on eigenvalue stability of Hermitian and normal matrices (arXiv: 2603.23056) establishes the precise regularity: ordered eigenvalues form a Lipschitz map — they cannot change faster than the matrix entries, at a controlled rate. Moreover, the eigenvalue map acts continuously on Sobolev spaces W^{1,q} for q < ∞ but not for q = ∞, where discontinuity appears.
The Lipschitz property means eigenvalue crossings (where two eigenvalues become equal) do not create derivative singularities strong enough to destroy integrability. But the failure at q = ∞ means that eigenvalue crossings CAN create discontinuities in the supremum norm — the worst-case pointwise behavior is singular even though the average behavior is tame.
For normal matrices (the non-Hermitian generalization where the matrix commutes with its adjoint), eigenvalues are complex-valued and the stability result extends through multivalued Sobolev functions — acknowledging that eigenvalues can swap labels at crossings.
The through-claim: eigenvalue stability has a sharp regularity boundary at L^∞. Eigenvalues are stable enough for any finite-exponent Sobolev space but not for the space of bounded derivatives. The crossings — where eigenvalues meet and repel — are exactly regular enough for integration but not for pointwise control. The boundary between stable and unstable is the passage from average to worst case.
2603.23056. Linear algebra / eigenvalue perturbation / Sobolev spaces / Lipschitz continuity / spectral theory.