Naslov
Optimal perturbation bounds for the Hermitian eigenvalue problem

Autori
Barlow Jesse ; Slapničar, Ivan

Izvornik
Linear Algebra and Its Applications (0024-3795) 309 (2000); 19-43

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

Ključne riječi
relative error; zero subspace; pseudoinverse; relative gap; absolute gap

Sažetak
There is now a large literature on structured perturbation bounds for eigenvalue problems of the form \[ H x = \lambda M x, \] where $H$ and $M$ are Hermitian. These results give relative error bounds on the $i$th eigenvalue, $\lambda_i$, of the form \[ \frac{|\lambda_i - \tilde{\lambda}_i |}{|\lambda_i|}, \] and bound the error in the $i$th eigenvector in terms of the relative gap, \[ \min_{j \neq i} \frac{|\lambda_i - \lambda_j |}{|\lambda_i \lambda_j|^{1/2}}. \] In general, this theory usually restricts $H$ to be nonsingular and $M$ to be positive definite. We relax this restriction by allowing $H$ to be singular. For our results on eigenvalues we allow $M$ to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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