Singular value perturbation bound ($|{\sigma }_j(A)-{\sigma }_j(B)|\le \Vert A-B{\Vert }_2$)

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 ${\sigma }_j(A)\le \Vert A-B_{j-1}{\Vert }_2\le \Vert A-B{\Vert }_2+{\sigma }_j(B)$. Interchanging $A,B$ proves the bound.