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-rw-r--r-- | vorlesungen/slides/5/zerlegung.tex | 105 |
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diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex new file mode 100644 index 0000000..a734d69 --- /dev/null +++ b/vorlesungen/slides/5/zerlegung.tex @@ -0,0 +1,105 @@ +% +% zerlegung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Zerlegung in Eigenräume} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.38] +\uncover<2->{ +\fill[color=blue!20] (0,11) rectangle (4,15); +\fill[color=red!20] (4,0) rectangle (15,11); +} +\uncover<3->{ +\fill[color=red!40] (9,0) rectangle (15,6); +\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11); +} +\uncover<4->{ +\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6); +\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3); +} + +\uncover<2->{ +\draw[line width=0.1pt] (0,11) -- (15,11); +\draw[line width=0.1pt] (4,0) -- (4,15); +} + +\uncover<3->{ +\draw[line width=0.1pt] (0,6) -- (15,6); +\draw[line width=0.1pt] (9,0) -- (9,15); +} + +\uncover<4->{ +\draw[line width=0.1pt] (0,3) -- (15,3); +\draw[line width=0.1pt] (12,0) -- (12,15); +} +\draw (0,0) rectangle (15,15); +\uncover<2->{ +\node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$}; +\node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$}; +\node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$}; +} +\uncover<3->{ +\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$}; +\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$}; +} +\uncover<4->{ +\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$}; +\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$}; +\node at (10.5,4.5) {$(f_2)_{|\mathcal{E}_{\lambda_3}}$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Iteration} +$\Lambda=\{\lambda_1,\dots,\lambda_s\}$ Eigenwerte +\begin{align*} +\uncover<2->{ +V +&= +\mathcal{K}(f-\lambda_1) +\oplus +\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!20] (0,0) rectangle (1.83,1.1);}}}} +\underbrace{\mathcal{J}(f-\lambda_1)}_{\displaystyle=V_1} +} +\\[-15pt] +\uncover<2->{ +f_1 &= f_{|V_1} +} +\\[10pt] +\uncover<3->{ +V_1 +&= +\mathcal{K}(f_1-\lambda_2) +\oplus +\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!40] (0,0) rectangle (1.9,1.1);}}}} +\underbrace{\mathcal{J}(f_1-\lambda_2)}_{\displaystyle=V_2} +} +\\[-15pt] +\uncover<3->{ +f_1 &= f_{|V_1} +} +\\ +\uncover<4->{ +&\phantom{0}\vdots +} +\end{align*} +\uncover<5->{% +$\Rightarrow$ $f$ hat {\color{blue}Blockdiagonalform} für die Zerlegung +\begin{align*} +V&=\bigoplus_{\lambda\in\Lambda} \mathcal{E}_{\lambda} +\end{align*}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |