Given a symmetric matrix $X \in \mathbb{R}^{n \times n}$. We know the following: trace$(X) = \sum_{i=1}^n x_{ii} = \sum_{i=1}^n \lambda_i$ where $x_{ii}$ is the $i$th element on the diagonal of $X$, and ${\lambda_i}$ are the eignevalues of $X$.
If we cannot guarantee that $X$ is positive-semidefinite, is there a way to write $\sum_{i=1}^n| \lambda_i |$ in terms of entries of $X$, and perhabs other slack variables?