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 by the method of the splitting operators. The asymptotic bounds are derived from the exact nonlinear expressions for the perturbations and make possible to bound each entry of every matrix eigenvector in case of distinct eigenvalues. In contrast to the perturbation analysis of the Schur form of a nonsymmetric matrix, the bounds obtained do not make use of the Kronecker product which reduces significantly the necessary memory and volume of computations. This allows to analyze efficiently the sensitivity of high order problems. The eigenvector perturbation bounds are applied to obtain bounds on the angles between the perturbed and unperturbed one-dimensional invariant subspaces spanned by the corresponding eigenvectors. To reduce the bound conservatism in case of high order problems, we propose to implement probabilistic perturbation bounds based on the Markoff inequality. The analysis is illustrated by two numerical experiments of order 5000.