Optimal perturbation bounds for the Hermitian eigenvalue problem

There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λ_i, of the form|λ_i-λ̃_i||λ_i|,and bound the error in the ith eigenvector in terms of the relative gap,min_j≠i|λ_i-λ _j||λ_iλ_j|^1/2. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.