Relative perturbation bound for invariant subspaces of graded indefinite Hermitian matrices

We give a bound for the perturbations of invariant subspaces of graded inde finite Hermitian matrix H = D* AD which is perturbed into H + delta H = D* (A + delta A)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 \\delta A\ \, and not the perturbation delta A itself. This typically occurs when data are given with relative uncertainties, when the matrix is being stored int o computer memory, and when analyzing some numerical algorithms. Subspace p erturbations are measured in terms of perturbations of angles between subsp aces, and our bound is therefore a relative variant of the well-known Davis -Kahan sin Theta theorem. Our bounds generalize some of the recent relative perturbation results. (C) 1999 Elsevier Science Inc. All rights reserved.