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

https://doi.org/10.1016/j.cam.2019.03.012Get rights and content
Under an Elsevier user license
open archive

Abstract

The main results of the paper are an upper and a lower bound for the Frobenius norm of the matrix sinΘ, of the sines of the canonical angles between unperturbed and perturbed eigenspaces of a regular generalized Hermitian eigenvalue problem Ax=λBx 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 λ2M+λD+Ku=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.

MSC

65F15
15A57
15A18
65F25
65F35
70Q05

Keywords

sinθ theorem
Perturbation theory
Generalized eigenvalue problem
Regular Hermitian matrix pairs
Damping
Mechanical systems

Cited by (0)