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
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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.

Domaines: Théorie spectrale [math.SP]; Combinatoire [math.CO]

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⟩