From 38f326c35243942285c43310751025d2c16801dd Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 4 Mar 2021 20:30:33 +0100
Subject: Spektraltheorie 1. Teil

---
 vorlesungen/slides/5/jordan.tex | 130 +++++++++++++++++++++++++++++++++++++++-
 1 file changed, 129 insertions(+), 1 deletion(-)

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

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
-- 
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