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 when some of these clustered eigenvalues or clustered singular values are perfectly relatively distinguishable from the rest. In this paper, we consider how eigenspaces of a Hermitian matrix A change when it is perturbed to e A = D AD and how singular values of a (nonsquare) matrix B change when it is perturbed to e B = D 1 BD 2 , where D, D 1 and D 2 are assumed to be close to identity matrices of suitable dimensions, or either D 1 or D 2 close to some unitary matrix. It is proved that under these kinds of perturbations, the change ..