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A Note on an Upper and a Lower Bound on Sines between Eigenspaces for regular Hermitian matrix pairs

The main results of the paper are un 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 $A x = \lambda B x$ where $A$ and $B$ are Hermitian $n \times n$ 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 $\left( \lambda^2 M + \lambda D + K \right) 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.

Sin Theta theorem ; Perturbation theory ; Generalized eigenvalue problem ; Regular Hermitian matrix pairs ; Damping ; Mechanical systems

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