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'Schur's inequality'
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| Title of object: |
Schur's inequality |
| Canonical Name: |
ShursInequality |
| Type: |
Theorem |
| Created on: |
2003-06-28 09:43:34 |
| Modified on: |
2006-06-12 02:18:06 |
| Classification: |
msc:15A42, msc:26D15 |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
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\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |} |
Content:
{\bf Theorem} (Shur'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} |
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