Relative Perturbation Bound for Invariant Subspaces of Hermitian Matrix (original) (raw)

Abstract. We give a bound for the perturbations of invariant sub-spaces of a non-singular Hermitian matrix H under relative additive per-turbations of H. Such perturbations include the case when the elements ofH are known up to some relative tolerance. Our bound is, in appropriatecases, sharper than the classical bounds, and it generalizes some of therecent relative perturbation results. 1. Introduction and preliminariesWe consider the Hermitian eigenvalue problemH= Q Q =X ni=1  i q i q ;where H is a non-singular Hermitian matrix of order n, = diag( i ) is adiagonal matrix whose diagonal elements are the eigenvalues of H, and Q=q 1 q 2 q n isanunitarymatrixwhosei-th columnisthe eigenvectorwhich corresponds to  i . We denote the set of all eigenvalues of Hby ˙(H) =f 1 ; ; n g. We also assume that the eigenvalues are ordered,  1   2    n .Subspace X is an invariant subspace of a general matrix Hif HX  X.We consider invariant subspaces which correspond to the set of kneighb...