A note on an upper and a lower bound on sines between eigenspaces for regular Hermitian matrix pairs

Abstract The main results of the paper are un upper and a lower bound for the Frobenius norm of the matrix sin Θ , of the sines of the canonical angles between unperturbed and perturbed eigenspaces of a regular generalized Hermitian eigenvalue problem A x = λ B x where A and B are Hermitian n × n matrices, under a feasible non-Hermitian perturbation. As one application of the obtained bounds we present the corresponding upper and the lower bounds for eigenspaces of a matrix pair ( A , B ) obtained by a linearization of regular quadratic eigenvalue problem λ 2 M + λ D + K u = 0 , where M is positive definite and D and K are semidefinite. We also apply obtained upper and lower bounds to the important problem which considers the influence of adding a damping on mechanical systems. The new results show that for certain additional damping the upper bound can be too pessimistic, but the lower bound can reflect a behaviour of considered eigenspaces properly. The obtained results have been illustrated with several numerical examples.