Relative Perturbation Theory: II. Eigenspace and Singular Subspace Variations

The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on invariant subspace variations that are proportional to the reciprocals of absolute gaps between subsets of spectra or subsets of singular values. These bounds may be bad news for invariant subspaces corresponding to clustered eigenvalues or clustered singular values of much smaller magnitudes than the norms of matrices under considerations. In this paper, we consider how eigenspaces of a Hermitian matrix A change when it is perturbed to $\widetilde A=D^*AD$ and how singular spaces of a (nonsquare) matrix B change when it is perturbed to $\widetilde B=D_1^*BD_2$, where D, D1, and D2 are nonsingular. It is proved that under these kinds of perturbations, the changes of invariant subspaces are proportional to the reciprocals of relative gaps between subsets of spectra or subsets of singular values. The classical Davis--Kahan $\sin\theta$ theorems and Wedin $\sin\theta$ theorems are extended.