Tropical bounds for eigenvalues of matrices

Tropical bounds for eigenvalues of matrices

Let λ1,…,λn denote the eigenvalues of a n×n matrix, ordered by nonincreasing absolute value, and let γ1≥⋯≥γn denote the tropical eigenvalues of an associated n×n matrix, obtained by replacing every entry of the original matrix by its absolute value. We show that for all 1≤k≤n, |λ1⋯λk|≤Cn,kγ1⋯γk, whe...

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Veröffentlicht in:Linear algebra and its applications 2014-04, Vol.446, p.281-303
Hauptverfasser: Akian, Marianne, Gaubert, Stéphane, Marchesini, Andrea
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Sprache:eng
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Combinatorics
Location of eigenvalues
Log-majorization
Mathematics
Ostrowskiʼs inequalities
Parametric optimal assignment
Spectral Theory
Tropical geometry
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Gaubert, Stéphane
Marchesini, Andrea
description Let λ1,…,λn denote the eigenvalues of a n×n matrix, ordered by nonincreasing absolute value, and let γ1≥⋯≥γn denote the tropical eigenvalues of an associated n×n matrix, obtained by replacing every entry of the original matrix by its absolute value. We show that for all 1≤k≤n, |λ1⋯λk|≤Cn,kγ1⋯γk, where Cn,k is 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.
doi_str_mv 10.1016/j.laa.2013.12.021
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subjects Combinatorics
Location of eigenvalues
Log-majorization
Mathematics
Ostrowskiʼs inequalities
Parametric optimal assignment
Spectral Theory
Tropical geometry
title Tropical bounds for eigenvalues of matrices
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