From f297ac1a4e95db255ff6c8e892904c1743ed57e5 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Wed, 3 Mar 2021 22:51:16 +0100
Subject: add new slides

---
 vorlesungen/slides/5/zerlegung.tex | 73 ++++++++++++++++++++++++++++++++++++++
 1 file changed, 73 insertions(+)
 create mode 100644 vorlesungen/slides/5/zerlegung.tex

(limited to 'vorlesungen/slides/5/zerlegung.tex')

diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex
new file mode 100644
index 0000000..9c20c60
--- /dev/null
+++ b/vorlesungen/slides/5/zerlegung.tex
@@ -0,0 +1,73 @@
+%
+% 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]
+\fill[color=blue!20] (0,11) rectangle (4,15);
+\fill[color=red!20] (4,0) rectangle (15,11);
+\fill[color=red!40] (9,0) rectangle (15,6);
+\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11);
+\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6);
+\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3);
+\draw[line width=0.1pt] (0,11) -- (15,11);
+\draw[line width=0.1pt] (0,6) -- (15,6);
+\draw[line width=0.1pt] (0,3) -- (15,3);
+\draw[line width=0.1pt] (4,0) -- (4,15);
+\draw[line width=0.1pt] (9,0) -- (9,15);
+\draw[line width=0.1pt] (12,0) -- (12,15);
+\draw (0,0) rectangle (15,15);
+\node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$};
+\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$};
+\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$};
+\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$};
+\node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$};
+\node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$};
+\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$};
+\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*}
+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]
+f_1 &= f_{|V_1}
+\\[10pt]
+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]
+f_1 &= f_{|V_1}
+\\
+&\phantom{0}\vdots
+\end{align*}
+$\Rightarrow$ $f$ 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
-- 
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