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| Formaat: | Recurso digital |
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Zenodo
2025
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| Online toegang: | https://doi.org/10.5281/zenodo.17689768 |
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Inhoudsopgave:
- This paper investigates the structural stability of various matrix canonical forms when subjected to small perturbations. Matrix canonical forms, such as the Jordan canonical form, Frobenius rational canonical form, and block diagonal forms, are fundamental tools in linear algebra for analyzing the intrinsic properties of linear operators and systems. However, their practical utility, particularly in numerical computations and real-world applications, is often challenged by their sensitivity to data inaccuracies or measurement noise. We delve into how infinitesimal changes in a matrix's entries can drastically alter its canonical representation, focusing specifically on the implications for eigenvalues, eigenvectors, and the block structure. The study systematically examines the conditions under which these forms exhibit robustness or extreme fragility, highlighting the critical role of eigenvalue multiplicity and geometric properties. We discuss classical perturbation theory results, including those by Wilkinson and Kato, and extend the analysis to modern concepts like the pseudo-spectrum, which offers a more realistic measure of stability in the presence of perturbations. The findings underscore the inherent challenges in numerically computing canonical forms and provide insights into the design of more robust algorithms for system analysis and control. The paper contributes to a deeper understanding of the theoretical underpinnings and practical limitations of using canonical forms in perturbed environments.