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Tropical bounds for eigenvalues of matrices

Authors
  • Akian, Marianne
  • Gaubert, Stéphane
  • Marchesini, Andrea1, 2, 3
  • 1 INRIA
  • 2 CMAP
  • 3 École Polytechnique
Type
Published Article
Journal
Linear Algebra and its Applications
Publisher
Elsevier
Publication Date
Jan 01, 2013
Accepted Date
Dec 11, 2013
Volume
446
Pages
281–303
Identifiers
DOI: 10.1016/j.laa.2013.12.021
Source
Elsevier
Keywords
License
Unknown

Abstract

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.

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