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Diffstat (limited to 'vorlesungen/slides/5/jordan.tex')
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1 files changed, 129 insertions, 1 deletions
diff --git a/vorlesungen/slides/5/jordan.tex b/vorlesungen/slides/5/jordan.tex index ad0e31d..e6ece47 100644 --- a/vorlesungen/slides/5/jordan.tex +++ b/vorlesungen/slides/5/jordan.tex @@ -3,8 +3,136 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\L#1{ + \node at ({#1-0.5},{0.5-#1}) {$\lambda$}; +} +\def\E#1{ + \node at ({#1-0.5},{1.5-#1}) {$1$}; +} + \begin{frame}[t] \frametitle{Jordan Normalform} -XXX +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Wahl der Basis} +\begin{enumerate} +\item<2-> Zerlegung in verallgemeinerte Eigenräume +\begin{align*} +V +&= +\mathcal{E}_{{\color{blue}\lambda}}(A) +\oplus +\mathcal{E}_{{\color{darkgreen}\lambda}}(A) +\oplus +\mathcal{E}_{{\color{red}\lambda}}(A) +%\oplus +%\dots +\\ +\llap{$A\mathcal{E}_{{\color{blue}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{blue}\lambda}}(A) +\\ +\llap{$A\mathcal{E}_{{\color{darkgreen}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{darkgreen}\lambda}}(A) +\\ +\llap{$A\mathcal{E}_{{\color{red}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{red}\lambda}}(A), +\dots +\end{align*} +\item<3-> In jedem Eigenraum: Zerlegung in Jordan-Blöcke +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.33] +\fill[color=gray!20] (0,-20) rectangle (20,0); +\node[color=white] at (10,-10) [scale=12] {$A$}; + +\uncover<2->{ + \fill[color=blue!20,opacity=0.5] (0,0) rectangle (8,-8); + \fill[color=darkgreen!20,opacity=0.5] (8,-8) rectangle (15,-15); + \fill[color=red!20,opacity=0.5] (15,-15) rectangle (20,-20); + \fill[color=blue!20] (0,0) rectangle (8,2); + \fill[color=blue!20] (-2,-8) rectangle (0,0); + \fill[color=darkgreen!20] (8,0) rectangle (15,2); + \fill[color=darkgreen!20] (-2,-15) rectangle (0,-8); + \fill[color=red!20] (15,0) rectangle (20,2); + \fill[color=red!20] (-2,-20) rectangle (0,-15); +} + +\uncover<3->{ + \draw[color=gray] (0,0) rectangle (5,-5); + \draw[color=gray] (5,-5) rectangle (8,-8); + \draw[color=gray] (8,-8) rectangle (15,-15); + \draw[color=gray] (15,-15) rectangle (16,-16); + \draw[color=gray] (16,-16) rectangle (17,-17); + \draw[color=gray] (17,-17) rectangle (20,-20); +} + +\uncover<2->{ + \draw[color=gray] (8,0) -- (8,-20); + \draw[color=gray] (15,0) -- (15,-20); + \draw[color=gray] (0,-8) -- (20,-8); + \draw[color=gray] (0,-15) -- (20,-15); + + \node at (0,-4) [above,rotate=90] + {$\mathcal{E}_{{\color{blue}\lambda}}(A)$}; + \node at (4,0) [above] + {$\mathcal{E}_{{\color{blue}\lambda}}(A)$}; + \node at (0,-11.5) [above,rotate=90] + {$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$}; + \node at (11.5,0) [above] + {$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$}; + \node at (0,-18.5) [above,rotate=90] + {$\mathcal{E}_{{\color{red}\lambda}}(A)$}; + \node at (18.5,0) [above] + {$\mathcal{E}_{{\color{red}\lambda}}(A)$}; +} +\uncover<2->{ + {\color{blue} + \foreach \x in {1,...,8}{ \L{\x} } + } + {\color{darkgreen} + \foreach \x in {9,...,15}{ \L{\x} } + } + {\color{red} + \foreach \x in {16,...,20}{ \L{\x} } + } +} + +\uncover<3->{ +\E{2} +\E{3} +\E{4} +\E{5} + +\E{7} +\E{8} + +\E{10} +\E{11} +\E{12} +\E{13} +\E{14} +\E{15} + +\E{19} +\E{20} +} + +\draw (0,-20) rectangle (20,0); +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} \end{frame} + +\egroup |