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 .
The present inequalities generalize a theorem of Friedland, who showed in @cite_2 that for a nonnegative matrix @math , we have where @math is the of @math , defined to be [ (A) = i_1, ,i_ ( a_ i_1,i_2 a_ i_2,i_3 a_ i_ ,i_1 ) ^ 1 ; , ] the maximum being taken over all sequences @math of distinct elements of @math . Since @math , the second inequality in corresponds to the case @math in .
@cite_2
2025314573
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.