singular-value perturbation

Singular values are the most robust numbers attached to a matrix. Unlike eigenvalues of a general matrix, which can be wildly sensitive, singular values barely budge under perturbation — even for non-symmetric, non-normal, or rectangular matrices. This is why the SVD is the workhorse of noisy numerical computation.

Weyl's bound for singular values says each one is 1-Lipschitz in the matrix: |sigma_k(A + E) - sigma_k(A)| <= ||E||_2 for every k, where the singular values are ordered sigma_1 >= sigma_2 >= ... Mirsky's stronger theorem bounds them collectively: sqrt(sum_k (sigma_k(A + E) - sigma_k(A))^2) <= ||E||_F, and more generally the change is bounded in every unitarily invariant norm.

The reason singular values are so stable is that sigma_k(A) are the (nonnegative square roots of the) eigenvalues of the symmetric matrix A^T A, and symmetric eigenvalues obey Weyl's perfectly conditioned bound. So singular values inherit the best-case behavior automatically, with no eigenvector-conditioning factor in sight.

This stability is the backbone of practical algorithms: numerical rank counts singular values above a tolerance precisely because noise of size epsilon can only move them by epsilon, and low-rank approximation is reliable because truncating small singular values discards a controlled amount of the matrix. Robustness to noise is the SVD's signature gift.