On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces

Abstract

The Rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices. The Rayleigh–Ritz method finds the stationary values of the Rayleigh quotient, called Ritz values, on a given trial subspace as optimal, in some sense, approximations to eigenvalues. In the present paper, we derive upper bounds for proximity of the Ritz values in terms of the proximity of the trial subspaces without making an assumption that the trial subspace is close to an invariant subspace. The main result is that the absolute value of the perturbations in the Ritz values is bounded by a constant times the gap between the original trial subspace and its perturbation. The constant is the spread in the matrix spectrum, i.e. the difference between the largest and the smallest eigenvalues of the matrix. It’s shown that the constant cannot be improved. We then generalize this result to arbitrary unitarily invariant norms, but we have to increase the constant by a factor of 2 . Our results demonstrate, in particular, the stability of the Ritz values with respect to a perturbation in the trial subspace.