Relative perturbation theory: (II) Eigenspace and singular subspace variations

摘要

The classical perturbation theory for Hermitian matrix enigenvalue and singular value problems provides bounds on invariant subspace variations that are proportional to the reciporcals 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 ro clustered singular values are perfectly relatively distinguishable from the rest. This paper considers how eigenvalues of a Hermitian matrix A change when it is perturbed to (tilde A)= D(sup (asterisk))AD and how singular values of a (nonsquare) matrix B change whenit is perturbed to (tilde B)=D(sub 1)(sup (asterisk))BD(sub 2), where D, D(sub 1), and D(sub 2) are assumed to be close to identity matrices of suitable dimensions, or either D(sub 1) or D(sub 2) close to some unitary matrix. It is proved that under these kinds of perturbations, the change of invarient subspaces are proportional to reciprocals of relative gaps between subsets of spectra or subsets of singular values. We have been able to extend well-known Davis-Kahan sin (theta) theorems and Wedin sin (theta) theorems. As applications, we obtained bounds for perturbations of graded matrices.