Off-diagonal perturbation, first-order approximation and quadratic residual bounds for matrix eigenvalue problems

Abstract

When a symmetric block diagonal matrix (Formula Presented) undergoes an off-diagonal perturbation (Formula Presented), the eigenvalues of these matrices are known to differ only by O(∥E12∥2/gap), which scales quadratically with the norm of the perturbation. Here gap measures the distance between eigenvalues, and plays a key role in the constant. Closely related is the first-order perturbation expansion for simple eigenvalues of a matrix. It turns out that the accuracy of the first-order approximation is also O(∥E∥2/gap), where E is the perturbation matrix. Also connected is the residual bounds of approximate eigenvalues obtained by the Rayleigh-Ritz process, whose accuracy again scales quadratically in the residual, and inverse-proportionally with the gap between eigenvalues. All these are tightly linked, but the connection appears to be rarely discussed. This work elucidates this connection by showing that all these results can be understood in a unifying manner via the quadratic perturbation bounds of block diagonal matrices undergoing off-diagonal perturbation. These results are essentially known for a wide range of eigenvalue problems: symmetric eigenproblems (for which the explicit constant can be derived), nonsymmetric and generalized eigenvalue problems. We also extend such results to matrix polynomials, and show that the accuracy of a first-order expansion also scales as O(∥E∥2/gap), and argue that two-sided projection methods are to be preferred to one-sided projection for nonsymmetric eigenproblems, to obtain higher accuracy in the computed eigenvalues.