Theorem (Schur's inequality) Let $A$ be a square $n\times n$ matrix with real (or possibly complex entries). If $\lambda_1,\ldots, \lambda_n$ are the eigenvalues of $A$ , and $D$ is the diagonal matrix $D=\operatorname{diag}(\lambda_1,\ldots, \lambda_n)$ , then \begin{eqnarray*} \Vert D \Vert_F &\le& \Vert A \Vert_F, \end{eqnarray*}where $\Vert\cdot \Vert_F$ is the Frobenius matrix norm. Equality holds if and only if $A$ is a normal matrix.

References

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V.V. Prasolov, Problems and Theorems in Linear Algebra, American Mathematical Society, 1994.