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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:358

Article

Truhar, Ninoslav¹ 

[1] JJ Strossmayer Univ Osijek, Dept Math, Trg Ljudevita Gaja 6, Osijek 31000, Croatia

关键词: sin theta theorem;    Perturbation theory;    Generalized eigenvalue problem;    Regular Hermitian matrix pairs;    Damping;    Mechanical systems;   

DOI  :  10.1016/j.cam.2019.03.012

来源: Elsevier

【 摘 要 】

The main results of the paper are an upper and a lower bound for the Frobenius norm of the matrix sin Theta, of the sines of the canonical angles between unperturbed and perturbed eigenspaces of a regular generalized Hermitian eigenvalue problem A chi = lambda B chi where A and B are Hermitian n x 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 (lambda M-2 + lambda 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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free