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Comptes Rendus Mathématique
Volume 339, n° 2
pages 103-108 (juillet 2004)
Doi : 10.1016/j.crma.2004.05.001
Received : 25 February 2004 ;  accepted : 4 May 2004
Perturbation of eigenvalues of matrix pencils and the optimal assignment problem

Marianne  Akian a ,  Ravindra  Bapat b ,  Stéphane  Gaubert a
aINRIA, domaine de Voluceau, B.P. 105, 78153 Le Chesnay cedex, France 
bIndian Statistical Institute, New Delhi, 110016, India 

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We extend the perturbation theory of Visik, Ljusternik and Lidskii to the case of eigenvalues of matrix pencils. This extension allows us to solve certain degenerate cases of this theory. We show that the first order asymptotics of the eigenvalues of a perturbed matrix pencil can be computed generically by methods of min-plus algebra and optimal assignment algorithms. We illustrate this result by discussing a singular perturbation problem considered by Najman. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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Nous étendons au cas des valeurs propres de faisceaux de matrices la théorie des perturbations de Visik, Ljusternik et Lidskii, ce qui permet de résoudre certains cas dégénérés de cette théorie. Nous montrons que les asymptotiques au premier ordre des valeurs propres d'un faisceau perturbé peuvent être calculées génériquement au moyen de méthodes de l'algèbre min-plus et d'algorithmes d'affectation optimale. Nous illustrons ce résultat en discutant un problème de perturbation singulière considéré par Najman. Pour citer cet article : M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).




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