  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.
