Relative perturbation bound for invariant subspaces of graded indefinite Hermitian matrices

Abstract

We give a bound for the perturbations of invariant subspaces of graded indefinite Hermitian matrix H= D * AD which is perturbed into H+ δH= D *( A+ δA) D. Such relative perturbations include an important case where H is given with an element-wise relative error. Application of our bounds requires only the knowledge of the size of relative perturbation ∥ δA∥, and not the perturbation δA itself. This typically occurs when data are given with relative uncertainties, when the matrix is being stored into computer memory, and when analyzing some numerical algorithms. Subspace perturbations are measured in terms of perturbations of angles between subspaces, and our bound is therefore a relative variant of the well-known Davis–Kahan sin Θ theorem. Our bounds generalize some of the recent relative perturbation results.