Weyl's inequalities (eigenvalues)

Suppose you know the eigenvalues of a self-adjoint operator A and then nudge it to A + B with B also self-adjoint. How far can the eigenvalues move? Weyl's inequalities answer this with sharp bounds, and they are the foundation of eigenvalue perturbation theory. They fall straight out of the Courant-Fischer min-max characterization, because that formula compares Rayleigh quotients of A and A + B over the same subspaces.

Order eigenvalues in decreasing order. The simplest Weyl bound says each eigenvalue moves by no more than the size of the perturbation: |lambda_k(A + B) - lambda_k(A)| <= ||B||, where ||B|| is the largest |eigenvalue| of B (the operator norm). More refined Weyl inequalities give interlacing-style bounds such as lambda_{i+j-1}(A + B) <= lambda_i(A) + lambda_j(B), which sandwich the spectrum of the sum between sums of the individual spectra. They tell you exactly how the eigenvalues of A + B interlace those of A and B.

These bounds matter because real matrices are never known exactly. Measurement noise, rounding, and modeling error all perturb an operator, and Weyl guarantees the eigenvalues are stable: a small symmetric perturbation can only move each eigenvalue a small amount. This Lipschitz stability is special to the self-adjoint world — non-normal matrices can have eigenvalues that jump wildly under tiny perturbations. The same circle of ideas yields the Cauchy interlacing theorem (deleting a row and column shifts each eigenvalue by at most one slot) and Lidskii-type majorization results.