Schur’s inequality

Theorem (Schur’s inequality) Let $A$ be a square $n\times n$ matrix with real (or possibly complex entries). If ${\lambda }_1,\mathrm{\dots },{\lambda }_n$ 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.

Title	Schur’s inequality
Canonical name	SchursInequality
Date of creation	2013-03-22 13:43:30
Last modified on	2013-03-22 13:43:30
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	14
Author	matte (1858)
Entry type	Theorem
Classification	msc 26D15
Classification	msc 15A42
Related topic	TraceOfAMatrix
Related topic	WielandtHoffmanTheorem
Related topic	FrobeniusMatrixNorm