Schur’s inequality

Theorem (Schur’s inequality)Let $A$ be a square $n\times n$ matrix withreal (or possibly complex entries).If ${\lambda }_1,\mathrm{\dots },{\lambda }_n$ are the eigenvalues of $A$, and$D$ is the diagonal matrix $D=\mathrm{diag}({\lambda }_1,\mathrm{\dots },{\lambda }_n)$,then

${\parallel D\parallel }_F$$\le$${\parallel A\parallel }_F,$

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