Tropical bounds for the eigenvalues of block structured matrices

Marianne Akian (1, 2) , Stephane Gaubert (1, 2) , Andrea Marchesini (2, 1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 MAXPLUS - Max-plus algebras and mathematics of decision

We establish a log-majorization inequality, which relates the moduli of the eigenvalues of a block structured matrix with the tropical eigenvalues of the matrix obtained by replacing every block entry of the original matrix by its norm. This inequality involves combinatorial constants depending on the size and pattern of the matrix. Its proof relies on diagonal scalings, constructed from the optimal dual variables of a parametric optimal assignment problem.

Domaines: Théorie spectrale [math.SP]

Marianne Akian, Stephane Gaubert, Andrea Marchesini. Tropical bounds for the eigenvalues of block structured matrices. SIAM Conference on Applied Linear Algebra (SIAM LA), Oct 2015, Atlanta, United States. ⟨hal-01252379⟩