Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically:

Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with trace equal to zero. If $\lambda_1,\dots,\lambda_n$ be all of eigenvalues of $A$, then $$\sum_i |\lambda_i|\geq \frac{1}{n-1} \sum_{i\neq j} \big\vert a_{ij}-\frac{a_{ii}+a_{jj}}{2}\big\vert.$$

I have a very long combinatorial proof for this inequality, but I'm curious if there is a more straightforward proof. Thanks!