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/normalnilp.tex | 229 ++++++++++++++++++++++++++++++++++++
 1 file changed, 229 insertions(+)

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

diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex
index a7af682..9457136 100644
--- a/vorlesungen/slides/5/normalnilp.tex
+++ b/vorlesungen/slides/5/normalnilp.tex
@@ -3,6 +3,235 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 %
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\sx{1.9}
+\def\sy{0.6}
+\def\punkt#1#2#3{
+	\foreach \y in {0,...,#2}{
+	}
+}
+\def\block#1#2{
+	\fill[rounded corners=2pt,color=white]
+		({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05});
+	\draw[rounded corners=2pt]
+		({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05});
+}
+\def\teilmenge#1#2#3{
+	\fill[rounded corners=2pt,color=white]
+		({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=gray]
+		({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\rot#1#2#3{
+	\fill[rounded corners=2pt,color=red!20]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=red]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\hellblau#1#2#3{
+	\fill[rounded corners=2pt,color=blue!20]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=blue!40]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\punkt#1#2{
+	\fill[color=blue] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08];
+}
+\def\bildpunkt#1#2{
+	\fill[color=blue!40] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08];
+}
+\def\pfeil#1#2#3{
+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.1cm] 
+	({-#1*\sx},{(#2-0.5)*\sy}) 
+	--
+	({-(#1-1)*\sx},{(#3-0.5)*\sy}) ;
+}
 \begin{frame}[t]
 \frametitle{Normalform einer nilpotenten Matrix}
+{\usebeamercolor[fg]{title}$A^l=0$ $\Rightarrow$ finde eine ``gute'' Basis}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\vspace{-25pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=darkgreen!20,rounded corners=2pt]
+	({-3*\sx+0.35},0) -- (-0.35,0)
+	--
+	({-1*\sx+0.35},{4*\sy}) -- ({-1*\sx-0.35},{4*\sy})
+	--
+	({-2*\sx+0.35},{7*\sy}) -- ({-2*\sx-0.35},{7*\sy})
+	--
+	({-3*\sx+0.35},{8*\sy}) -- cycle;
+
+\block{0}{0}
+
+\block{1}{4}
+\uncover<10->{
+	\rot{1}{0}{1}
+	\node[color=red] at ({-1*\sx-0.28},{0.5*\sy}) [left] {$\mathcal{C}_{l-2}$};
+}
+\uncover<8->{
+	\hellblau{1}{1}{3}
+}
+\uncover<4->{
+	\hellblau{1}{3}{4}
+}
+
+\block{2}{7}
+\uncover<4->{
+	\hellblau{2}{6}{7}
+}
+\uncover<6->{
+	\rot{2}{4}{6}
+	\node[color=red] at ({-2*\sx-0.28},{5*\sy}) [left] {$\mathcal{C}_{l-1}$};
+}
+\teilmenge{2}{0}{4}
+
+\block{3}{8}
+\uncover<2->{
+	\rot{3}{7}{8}
+	\node[color=red] at ({-3*\sx-0.28},{7.5*\sy}) [left] {$\mathcal{C}_l$};
+}
+\teilmenge{3}{0}{7}
+
+\uncover<3->{
+	\punkt{3}{8}
+}
+\uncover<4->{
+	\pfeil{3}{8}{7}
+	\bildpunkt{2}{7}
+	\pfeil{2}{7}{4}
+	\bildpunkt{1}{4}
+}
+
+\uncover<7->{
+	\punkt{2}{5}
+	\punkt{2}{6}
+}
+\uncover<8->{
+	\pfeil{2}{5}{2}
+	\bildpunkt{1}{3}
+	\pfeil{2}{6}{3}
+	\bildpunkt{1}{2}
+}
+
+\uncover<11->{
+\punkt{1}{1}
+}
+
+\node at ({-3*\sx},0) [below] {$\mathcal{K}^l(A)\mathstrut$};
+\node at ({-2*\sx},0) [below] {$\mathcal{K}^{l-1}(A)\mathstrut$};
+\node at ({-1.45*\sx},0) [below] {$\dots\mathstrut$};
+\node at ({-1*\sx},0) [below] {$\mathcal{K}^1(A)\mathstrut$};
+\node at ({-0*\sx},0) [below] {$0=\mathcal{K}^0(A)\mathstrut$};
+\node[color=gray] at ({-2*\sx},{2*\sy}) [rotate=90] {$\mathcal{K}^1(A)$};
+\node[color=gray] at ({-3*\sx},{3.5*\sy}) [rotate=90] {$\mathcal{K}^{l-1}(A)$};
+\foreach \x in {0,1,2}{
+	\draw[->,shorten >= 0.1cm, shorten <= 0.1cm]
+		({-(\x+1)*\sx},{8.7*\sy}) -- ({-(\x)*\sx},{8.7*\sy});
+	\node at ({-(\x+0.5)*\sx},{8.7*\sy}) [above] {$A$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\vspace{-30pt}
+\begin{enumerate}
+\item<2->	\(
+	\mathcal{K}^l(A)=\mathcal{K}^{l-1}\oplus {\color{red}\mathcal{C}_l}
+	\)
+\item<3->	\(
+	{\color{blue}b_1}\in{\color{red}\mathcal{C}_l}
+	\)
+\item<4->	\(
+	\mathcal{B}_l
+	=
+	\{{\color{blue}b_1},{\color{blue!40}Ab_1},{\color{blue!40}A^2b_1},\dots,
+	{\color{blue!40}A^{l-1}b_1}\}
+	\)
+\item<5->	\(
+	\mathcal{K}^{l-1}(A)
+	=
+	\mathcal{K}^{l-2}(A)
+	\oplus
+	{\color{red}\mathcal{C}_{l-1}}
+	\oplus
+	{\color{blue}A\mathcal{C}_l}
+	\)
+\item<6->	\(
+	{\color{blue}b_2},{\color{blue}b_3}\in{\color{red}\mathcal{C}_{l-1}}
+	\)
+\item<7->	\(
+	\mathcal{B}_{l-1}
+	=
+	\{
+	{\color{blue}b_2},{\color{blue}b_3},
+	{\color{blue!40}Ab_2}, {\color{blue!40}Ab_3},\dots
+	\}
+	\)
+\item<8-> \dots
+\end{enumerate}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+
+\uncover<2-4>{
+	\fill[color=red!20] (2,0) rectangle (3,8);
+}
+\uncover<4->{
+	\fill[color=blue!20] (0,6) rectangle (2,8);
+}
+\uncover<5->{
+	\fill[color=red!20] (2,5) rectangle (3,8);
+	\node[color=blue] at (2.5,6.5) {$1$};
+	\node[color=blue] at (1.5,7.5) {$1$};
+	\node[color=gray] at (0.5,7.5) {$0$};
+	\node[color=gray] at (1.5,6.5) {$0$};
+	\node[color=gray] at (2.5,5.5) {$0$};
+	\draw[color=gray] (0.05,5.05) rectangle (2.95,7.95);
+}
+
+\uncover<6-8>{
+	\fill[color=red!20] (4,0) rectangle (5,8);
+	\fill[color=red!20] (6,0) rectangle (7,8);
+}
+\uncover<8->{
+	\fill[color=blue!20] (3,4) rectangle (4,5);
+	\fill[color=blue!20] (5,2) rectangle (6,3);
+}
+\uncover<9->{
+	\fill[color=red!20] (4,3) rectangle (5,5);
+	\node[color=blue] at (4.5,4.5) {$1$};
+	\node[color=gray] at (3.5,4.5) {$0$};
+	\node[color=gray] at (4.5,3.5) {$0$};
+	\draw[color=gray] (3.05,3.05) rectangle (4.95,4.95);
+	\fill[color=red!20] (6,1) rectangle (7,3);
+	\node[color=blue] at (6.5,2.5) {$1$};
+	\node[color=gray] at (5.5,2.5) {$0$};
+	\node[color=gray] at (6.5,1.5) {$0$};
+	\draw[color=gray] (5.05,1.05) rectangle (6.95,2.95);
+}
+
+\uncover<10>{
+	\fill[color=red!20] (7,0) rectangle (8,8);
+}
+\uncover<11->{
+	\fill[color=red!20] (7,0) rectangle (8,1);
+	\node[color=gray] at (7.5,0.5) {$0$};
+	\draw[color=gray] (7.05,0.05) rectangle (7.95,0.95);
+}
+
+\draw (0,0) rectangle (8,8);
+\node at (-0.1,4) [left] {$A=$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
 \end{frame}
+\egroup
-- 
cgit v1.2.1

