Norm of a matrix and sum of eigenvalues

[Tolaso J Kos]

Let $A \in \mathcal{M}_{n \times n}(\mathbb{C})$ and let $\ell_1, \dots, \ell_n$ be its eigenvalues. Prove that:

$$\left \| A \right \|_2^2 \geq \sum_{i=1}^{n} \left| \ell_i \right|^2$$

where $\left \| \cdot \right \|_2$ denotes the $\displaystyle \left \| A \right \|_2 = \sqrt{\sum_{i=1}^{n}\sum_{j=1}^{n}\left | a_{ij} \right |^2}$ norm.

Edit: A typo and the declaration of the norm was corrected. To this new version I don't have a starting point as I had to the old version.