Wielandt-Hoffman theorem

Let $A$ and $B$ be normal matrices (http://planetmath.org/NormalMatrix). Let theireigenvalues $a_i$ and $b_i$ be ordered such that ${\sum }_i{|a_i-b_i|}^2$ isminimized. Then we have the following inequality

$$\sum _i{|a_i-b_i|}^2\le {\parallel A-B\parallel }_F^2,$$

where $\parallel \cdot {\parallel }_F$ is the Frobenius matrix norm.