A new method is introduced for deriving bounds on the relative change in the singular values of a real matrix due to a perturbation, as well as bounds on the angles between the unperturbed and perturbed singular vectors (or eigenvectors). The class of perturbations considered consists of all \(\delta B\) for which \(B+ \delta B= D_L BD_R\) for some nonsingular matrices \(D_L\) and \(D_R\). Many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations. Also, some new relative perturbation and deflation results for the singular values and vectors of biacyclic, triangular and shifted triangular matrices are presented.