Rigorous Asymptotic Perturbation Bounds for Hermitian Matrix Eigendecompositions

In this paper, we present rigorous asymptotic componentwise perturbation bounds for regular Hermitian indefinite matrix eigendecompositions, obtained via the method of splitting operators. The asymptotic bounds are derived from exact nonlinear expressions for the perturbations and allow each entry of every matrix eigenvector to be bounded in the case of distinct eigenvalues. In contrast to the perturbation analysis of the Schur form of a nonsymmetric matrix, the bounds obtained here do not rely on the Kronecker product, which significantly reduces both memory requirements and computational cost. This enables efficient sensitivity analysis of high-order problems. The eigenvector perturbation bounds are further applied to estimate the angles between perturbed and unperturbed one-dimensional invariant subspaces spanned by the corresponding eigenvectors. To reduce conservatism in the case of high-order problems, we propose the use of probabilistic perturbation bounds based on the Markov inequality. The analysis is illustrated by two numerical experiments of order 5000.