Schur's inequalityShursInequality2003-06-28 09:43:342006-06-12 02:20:26TheoremFrobenius matrix norm% this is the default PlanetMath preamble. as your knowledge
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\newcommand*{\abs}[1]{| #1 |}{\bf 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.
\begin{thebibliography}{9}
\bibitem{prasolov} V.V. Prasolov,
\emph{Problems and Theorems in Linear Algebra},
American Mathematical Society, 1994.
\end{thebibliography}