Schur's Inequalities

Let ${\hbox{\sf A}}=a_{ij}$ be an $n\times n$ Matrix with Complex (or Real) entries and Eigenvalues $\lambda_1$, $\lambda_2$, ..., $\lambda_n$, then

$$\sum_{i=1}^n \vert\lambda_i\vert^2\leq \sum_{i,j=1}^n \vert a_{ij}\vert^2$$

$$\sum_{i=1}^n \vert\Re[\lambda_i]\vert^2\leq \sum_{i,j=1}^n \left\vert{a_{ij}+a_{ji}^*\over 2}\right\vert^2$$

$$\sum_{i=1}^n \vert\Im[\lambda_i]\vert^2\leq \sum_{i,j=1}^n \left\vert{a_{ij}-a_{ji}^*\over 2}\right\vert^2.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1120, 1979.