Abstract Let λ 1 , … , λ n denote the eigenvalues of a n × n matrix, ordered by nonincreasing absolute value, and let γ 1 ≥ ⋯ ≥ γ n 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 , | λ 1 ⋯ λ k | ≤ C n , k γ 1 ⋯ γ k , where C n , 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 .
Abstract In this paper we give a simple closed formula for a certain limit eigenvalue of a nonnegative matrix. We bound the spectral radius of a nonnegative matrix in terms of this limit eigenvalue.