Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems (CROSBI ID 270842)
Prilog u časopisu | izvorni znanstveni rad | međunarodna recenzija
Podaci o odgovornosti
Truhar, Ninoslav ; Tomljanović, Zoran ; Li, Ren- Cang
engleski
Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems
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.
Quadratic matrix eigenvalue problem ; Perturbation theory ; Sin \Theta theorem ; Damped mechanical system
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Podaci o izdanju
371 (15)
2020.
124921
17
objavljeno
0096-3003
1873-5649
10.1016/j.amc.2019.124921