Combined Perturbation Bounds: I. Eigensystems and Singular Value Decompositions

In this paper we present some new combined perturbation bounds of eigenvalues and eigensubspaces for a Hermitian matrix $H$, particularly in an asymptotic sense, $\delta _{12}^{2}\Vert \sin \Theta (U_{1},\widetilde{U}_{1})\Vert _{F}^{2}+\sum_{i=1}^{r}(\lambda _{i}-\widetilde{\lambda }_{i})^{2} \le \Vert \Delta HU_{1}\Vert _{F}^{2}+O(\Vert \Delta HU_{1}\Vert _{F}^{4})$, where $\lambda _{i}$ denotes the eigenvalues of $H$ and $U_{1}$ the eigensubspace corresponding to the eigenvalues $\lambda _{i}$, $i=1,2,\ldots,r$. The bound for each factor of eigensystems is optimal due to the ${\rm{sin}}\Theta $ theorem and the Hoffman-Wielandt theorem. In addition, combined perturbation bounds for singular value decompositions and combined perturbation bounds in some, more general, measures are also obtained.