Perturbation theory for Hermitian quadratic eigenvalue problem - damped and simultaneously diagonalizable systems

Abstract The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems ( λ 2 M + λ D + K ) x = 0 . We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference | ∥ X 2 * M X ˜ 1 ∥ F 2 − ∥ X 2 * M X 1 ∥ F 2 | , and for ∥ X 2 * M X ˜ 1 − X 2 * M X 1 ∥ F , where the columns of X 1 = [ x 1 , … , x k ] and X 2 = [ x k + 1 , … , x n ] are linearly independent right eigenvectors and M is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for ∥ sin Θ ( X 1 , X ˜ 1 ) ∥ F , where Θ is a matrix of canonical angles between the eigensubspaces X 1 and X ˜ 1 , X 1 is spanned by the set of linearly independent right eigenvectors of SDQEP and X ˜ 1 is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.