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 (λ2M+λD+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 |∥X2*MX˜1∥F2−∥X2*MX1∥F2|, and for ∥X2*MX˜1−X2*MX1∥F, where the columns of X1=[x1,…,xk] and X2=[xk+1,…,xn] 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 ∥sinΘ(X1,X˜1)∥F, where Θ is a matrix of canonical angles between the eigensubspaces X1 and X˜1,X1 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.