Let $A = \left( {a_{ij} } \right)$ be a matrix over $\mathbb R$, of size $n \times n$. Let $\left\{ \lambda _k \right\}_{k = 1}^n$ be the $n$ eigenvalues of the matrix. Prove the following inequality:
$ \sum_{k = 1}^n \left| {\lambda _k } \right|^2 \leqslant \sum_{i = 1}^n \sum\limits_{j = 1}^n \left| {a_{ij} } \right|^2 . $
I can't prove it, and I'm not sure if in the problem one assumes that there exist $n$ real eigenvalues, or it's also true for complex eigenvalues. D: