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Tropical bounds for eigenvalues of matrices

Marianne Akian 1, 2 Stéphane Gaubert 1, 2 Andrea Marchesini 2, 1 
1 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : We show that for all k = 1,...,n the absolute value of the product of the k largest eigenvalues of an n-by-n matrix A is bounded from above by the product of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute value), up to a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1.
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Submitted on : Thursday, November 7, 2013 - 5:12:10 PM
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Marianne Akian, Stéphane Gaubert, Andrea Marchesini. Tropical bounds for eigenvalues of matrices. Linear Algebra and its Applications, 2014, 446, pp.281-303. ⟨10.1016/j.laa.2013.12.021⟩. ⟨hal-00881205⟩



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