A Computable Spectral Theorem

M. Ziegler, V. Brattka, in: Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000), Berlin, Heidelberg, 2001, pp. 378–388.

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Conference Paper | Published | English
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Abstract
Computing the spectral decomposition of a normal matrix is among the most frequent tasks to numerical mathematics. A vast range of methods are employed to do so, but all of them suffer from instabilities when applied to degenerate matrices, i.e., those having multiple eigenvalues. We investigate the spectral representation's effectivity properties on the sound formal basis of computable analysis. It turns out that in general the eigenvectors cannot be computed from a given matrix. If however the size of the matrix' spectrum (=number of different eigenvalues) is known in advance, it can be diagonalized effectively. Thus, in principle the spectral decomposition can be computed under remarkably weak non-degeneracy conditions.
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Proceedings Title
Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA'2000)
Volume
2064
Page
378-388
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Ziegler M, Brattka V. A Computable Spectral Theorem. In: Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000). Vol 2064. Berlin, Heidelberg; 2001:378-388. doi:10.1007/3-540-45335-0_23
Ziegler, M., & Brattka, V. (2001). A Computable Spectral Theorem. In Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000) (Vol. 2064, pp. 378–388). Berlin, Heidelberg. https://doi.org/10.1007/3-540-45335-0_23
@inproceedings{Ziegler_Brattka_2001, place={Berlin, Heidelberg}, title={A Computable Spectral Theorem}, volume={2064}, DOI={10.1007/3-540-45335-0_23}, booktitle={Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000)}, author={Ziegler, Martin and Brattka, Vasco}, year={2001}, pages={378–388} }
Ziegler, Martin, and Vasco Brattka. “A Computable Spectral Theorem.” In Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000), 2064:378–88. Berlin, Heidelberg, 2001. https://doi.org/10.1007/3-540-45335-0_23.
M. Ziegler and V. Brattka, “A Computable Spectral Theorem,” in Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000), 2001, vol. 2064, pp. 378–388.
Ziegler, Martin, and Vasco Brattka. “A Computable Spectral Theorem.” Proceedings of the 4th Workshop on Computability and Complexity in Analysis (CCA’2000), vol. 2064, 2001, pp. 378–88, doi:10.1007/3-540-45335-0_23.

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