Majorization inequalities for valuations of eigenvalues using tropical algebra

Marianne Akian 1, 2
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : We consider a matrix with entries over the field of Puiseux series, equipped with its non-archimedean valuation (the leading exponent). We establish majorization inequalities relating the sequence of the valuations of the eigenvalues of a matrix with the tropical eigenvalues of its valuation matrix (the latter is obtained by taking the valuation entrywise). We also show that, generically in the leading coefficients of the Puiseux series, the precise asymptotics of eigenvalues, eigenvectors and condition numbers can be determined. For this, we apply diagonal scalings constructed from the dual variables of a parametric optimal assignment constructed from the valuation matrix. Next, we establish an archimedean analogue of the above inequalities, which applies to matrix polynomials with coefficients in the field of complex numbers, equipped with the modulus as its valuation. This talk covers joint works with Ravindra Bapat, Stéphane Gaubert, Andrea Marchesini, and Francoise Tisseur.
Type de document :
Communication dans un congrès
4th International Conference on Matrix methods in Mathematics and Applications (MMMA-2015), Aug 2015, Moscow, Russia. 〈http://matrix.inm.ras.ru/mmma-2015/〉
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https://hal.inria.fr/hal-01252363
Contributeur : Marianne Akian <>
Soumis le : jeudi 7 janvier 2016 - 14:53:16
Dernière modification le : jeudi 10 mai 2018 - 02:05:35

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  • HAL Id : hal-01252363, version 1

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Marianne Akian. Majorization inequalities for valuations of eigenvalues using tropical algebra. 4th International Conference on Matrix methods in Mathematics and Applications (MMMA-2015), Aug 2015, Moscow, Russia. 〈http://matrix.inm.ras.ru/mmma-2015/〉. 〈hal-01252363〉

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