Ren Cang Li

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

Relative Perturbation Theory: I. Eigenvalue and Singular Value Variations

\widetilde A=D^*AD

Backward Error of Polynomial Eigenproblems Solved by Linearization

and how singular spaces of a (nonsquare) matrix B change when it is perturbed to

Composition constants for raising the orders of unconventional schemes for ordinary differential equations

\widetilde B=D_1^*BD_2

Alternating-directional Doubling Algorithm for

, 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

M

\sin\theta

-Matrix Algebraic Riccati Equations

theorems and Wedin

A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares Problems

\sin\theta

On perturbations of matrix pencils with real spectra

theorems are extended.

A perturbation bound for the generalized polar decomposition

The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on the absolute differences between approximate eigenvalues (singular values) and the true eigenvalues (singular values) of a matrix. These bounds may be bad news for small eigenvalues (singular values), which thereby suffer worse relative uncertainty than large ones. However, there are situations where even small eigenvalues are determined to high relative accuracy by the data much more accurately than the classical perturbation theory would indicate. In this paper, we study how eigenvalues of a Hermitian matrix A change when it is perturbed to