Naslov
Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems

Autori
Truhar, Ninoslav ; Tomljanović, Zoran ; Li, Ren- Cang

Izvornik
Applied mathematics and computation (0096-3003) 371 (2020), 15; 124921, 17

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Quadratic matrix eigenvalue problem ; Perturbation theory ; Sin \Theta theorem ; Damped mechanical system

Sažetak
The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems $(\lambda^2 M + \lambda D + K) x=0$. 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 $\left| \| X_2^* M \widetilde{; ; ; ; ; X}; ; ; ; ; _1 \|_F^2 - \| X_2^* M {; ; ; ; ; X}; ; ; ; ; _1 \|_F^2 \right|$, and for $\| X_2^* M \widetilde X_1 - X_2^* M X_1\|_F$, where the columns of $X_1=[x_1, \ldots, x_k]$ and $X_2= [x_{; ; ; ; ; k+1}; ; ; ; ; , \ldots, x_n]$ 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 $\|\sin{; ; ; ; ; \Theta(\mathcal{; ; ; ; ; X}; ; ; ; ; _1, \widetilde{; ; ; ; ; \mathcal{; ; ; ; ; X}; ; ; ; ; }; ; ; ; ; _1)}; ; ; ; ; \|_F$, where $\Theta$ is a matrix of canonical angles between the eigensubspaces $\mathcal{; ; ; ; ; X}; ; ; ; ; _1 $ and $\widetilde{; ; ; ; ; \mathcal{; ; ; ; ; X}; ; ; ; ; }; ; ; ; ; _1$, $\mathcal{; ; ; ; ; X}; ; ; ; ; _1 $ is spanned by the set of linearly independent right eigenvectors of SDQEP and $\widetilde{; ; ; ; ; \mathcal{; ; ; ; ; X}; ; ; ; ; }; ; ; ; ; _1$ is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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