I was asked by a student that
Given a matrix $(a_{ij})$ with its eigenvalues (with their multiplicities) $\{\lambda_i\}_{i=1}^n$, show that $$\sum_{i=1}^n|\lambda_i|\leq \sum_{i,j}|a_{ij}|$$
Frankly speaking, I am not sure that it is true, but I cannot find any counterexample by try some easy example.
