Tropical bounds for eigenvalues of matrices

Let λ1,…,λnλ1,…,λn denote the eigenvalues of a n×nn×n matrix, ordered by nonincreasing absolute value, and let γ1≥⋯≥γnγ1≥⋯≥γn denote the tropical eigenvalues of an associated n×nn×n matrix, obtained by replacing every entry of the original matrix by its absolute value. We show that for all 1≤k≤n1≤k≤n, |λ1⋯λk|≤Cn,kγ1⋯γk|λ1⋯λk|≤Cn,kγ1⋯γk, where Cn,kCn,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=1k=1.