Let $A$ and $B$ be normal matrices. Let their eigenvalues$a_i$ and $b_i$ be ordered such that $\sum_i |a_i-b_i|^2$ is minimized. Then we have the following inequality$$ \sum_i |a_i-b_i|^2 \le \|A-B\|_F^2,$$ where $\|\cdot\|_F$ is the Frobenius matrix norm.
