Singular value perturbation bound
= Singular value perturbation bound
{title2=$|\sigma_j(A)-\sigma_j(B)|\leq\|A-B\|_2$}
For equally sized <matrices>, corresponding ordered <singular values> differ by at most the <spectral norm> of the perturbation. Indeed, let $B_{j-1}$ be a best approximation of <matrix rank> at most $j-1$. The <Eckart-Young-Mirsky theorem> and the <triangle inequality> give $\sigma_j(A)\leq\|A-B_{j-1}\|_2\leq\|A-B\|_2+\sigma_j(B)$. Interchanging $A,B$ proves the bound.