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

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  • Truhar, Ninoslav
  • Tomljanović, Zoran
  • Li, Ren-Cang

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.

Suggested Citation

  • Truhar, Ninoslav & Tomljanović, Zoran & Li, Ren-Cang, 2020. "Perturbation theory for Hermitian quadratic eigenvalue problem – damped and simultaneously diagonalizable systems," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    • Handle: RePEc:eee:apmaco:v:371:y:2020:i:c:s0096300319309130
    DOI: 10.1016/j.amc.2019.124921
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