# Schur’s inequality

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

 $\displaystyle\|D\|_{F}$ $\displaystyle\leq$ $\displaystyle\|A\|_{F},$

where $\|\cdot\|_{F}$ is the Frobenius matrix norm. Equality holds if and only if $A$ is a normal matrix.

## References

Title Schur’s inequality SchursInequality 2013-03-22 13:43:30 2013-03-22 13:43:30 matte (1858) matte (1858) 14 matte (1858) Theorem msc 26D15 msc 15A42 TraceOfAMatrix WielandtHoffmanTheorem FrobeniusMatrixNorm