add slides

9 files changed, 566 insertions, 0 deletions

diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
index d690514..d081b29 100644
--- a/vorlesungen/slides/5/Makefile.inc
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -5,5 +5,12 @@
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
 chapter5 =								\
+	../slides/5/motivation.tex					\
+	../slides/5/charpoly.tex					\
+	../slides/5/kernbild.tex					\
+	../slides/5/ketten.tex						\
+	../slides/5/dimension.tex					\
+	../slides/5/folgerungen.tex					\
+	../slides/5/nilpotent.tex					\
 	../slides/5/chapter.tex
 
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
index 884732f..4bcee8e 100644
--- a/vorlesungen/slides/5/chapter.tex
+++ b/vorlesungen/slides/5/chapter.tex
@@ -3,3 +3,10 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
 %
+folie{5/motivation.tex}
+folie{5/charpoly.tex}
+folie{5/kernbild.tex}
+folie{5/ketten.tex}
+folie{5/dimension.tex}
+folie{5/folgerungen.tex}
+folie{5/nilpotent.tex}
diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex
new file mode 100644
index 0000000..1211b43
--- /dev/null
+++ b/vorlesungen/slides/5/charpoly.tex
@@ -0,0 +1,68 @@
+%
+% charpoly.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Charakteristisches Polynom über $\mathbb{C}$}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenwerte}
+Nur diejenigen $\mu$ kommen in Frage, für die
+$A-\mu I$ singulär ist:
+\[
+\chi_{A}(\mu)
+=
+\det (A-\mu I) = 0
+\]
+$\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$
+\end{block}
+\begin{block}{Zerlegung in Linearfaktoren}
+$\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$:
+\[
+\chi_A(X)
+=
+(X-\mu_1)\dots (X-\mu_n)
+\]
+\end{block}
+\begin{block}{Fundamentalsatz der Algebra}
+Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in
+Linearfaktoren
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Minimalpolynom}
+Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen
+\end{block}
+\begin{proof}[Beweis]
+\begin{enumerate}
+\item
+$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$
+\item
+$A-\lambda I$ ist regulär
+\end{enumerate}
+\begin{align*}
+&\Rightarrow&
+m_A(A)&=0
+\\
+&&
+(A-\lambda)^{-1}m_A(A) &=0
+\\
+&&
+\prod_{i\in I}(A-\mu_i)&=0,
+\end{align*}
+d.~h.~\(
+\displaystyle
+\overline{m}_A(X)
+=
+\prod_i{i\in I}(X-\mu_i)
+\in
+\mathbb{C}[X]
+\)
+\end{proof}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/dimension.tex b/vorlesungen/slides/5/dimension.tex
new file mode 100644
index 0000000..ff687b3
--- /dev/null
+++ b/vorlesungen/slides/5/dimension.tex
@@ -0,0 +1,68 @@
+%
+% dimension.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Dimension von $\mathcal{K}^k(f)$ und $\mathcal{J}^k(f)$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\pfad{
+	(0,0) -- (1,0.3) -- (2,0.9) 
+	--
+	(4,2.4) -- (5,2.7) -- (6,3.3)
+	--
+	(8,3.7) -- (9,4) -- (10,4) -- (11,4) -- (12,4)
+}
+
+\fill[color=darkgreen!20] \pfad -- (12,0) -- cycle;
+\fill[color=orange!20] \pfad -- (12,6) -- (0,6) -- cycle;
+
+\fill[color=darkgreen!40] (9,0) -- (12,0) -- (12,4) -- (9,4) -- cycle;
+\fill[color=orange!40] (9,4) -- (12,4) -- (12,6) -- (9,6) -- cycle;
+
+\node[color=orange] at (10.5,5) {$\mathcal{J}(f)$};
+\node[color=darkgreen] at (10.5,2) {$\mathcal{K}(f)$};
+
+\node[color=orange] at (5.5,4.5) {$\mathcal{J}^k(f)\supset\mathcal{J}^{k+1}(f)$};
+\node[color=darkgreen] at (5.5,1.5) {$\mathcal{K}^k(f)\subset\mathcal{K}^{k+1}(f)$};
+
+\draw[line width=1.4pt] \pfad;
+
+\draw[->] (-0.1,6) -- (12.5,6) coordinate[label={$k$}];
+\draw[->] (-0.1,0) -- (12.5,0) coordinate[label={$k$}];
+\node at (-0.1,6) [left] {$n$};
+\node at (-0.1,0) [left] {$0$};
+\foreach \x in {0,1,2,4,5,6,8,9,10,11,12}{
+	\fill (\x,0) circle[radius=0.05];
+	\fill (\x,6) circle[radius=0.05];
+}
+\node at (0,0) [below] {$0$};
+\node at (1,0) [below] {$1$};
+\node at (2,0) [below] {$2$};
+
+\node at (4,0) [below] {$k-1$};
+\node at (5,0) [below] {$k$};
+\node at (6,0) [below] {$k+1$};
+
+\node at (8,0) [below] {$l-1$};
+\node at (9,0) [below] {$l$};
+\node at (10,0) [below] {$l+1$};
+\node at (11,0) [below] {$l+2$};
+\node at (12,0) [below] {$l+3$};
+
+\fill (9,4) circle[radius=0.05];
+
+\node[color=orange] at (-0.2,3) [rotate=90] {$\dim\mathcal{J}^k(f)$};
+\node[color=darkgreen] at (12.2,2) [rotate=-90] {$\dim\mathcal{K}^k(f)$};
+
+\node[color=orange] at (9,5) [rotate=-90] {$\dim\mathcal{J}(f)$};
+\node[color=darkgreen] at (9,2) [rotate=-90] {$\dim\mathcal{K}(f)$};
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex
new file mode 100644
index 0000000..96efd7f
--- /dev/null
+++ b/vorlesungen/slides/5/folgerungen.tex
@@ -0,0 +1,31 @@
+%
+% folgerungen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Folgerungen}
+\begin{columns}[t]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zunahme}
+Für alle $k<l$ gilt
+\begin{align*}
+\mathcal{J}^k(f) &\supsetneq \mathcal{J}^{k+1}(f)
+\\
+\mathcal{K}^k(f) &\subsetneq \mathcal{K}^{k+1}(f)
+\end{align*}
+Für $k\ge l$ gilt
+\begin{align*}
+\mathcal{J}^k(f) &= \mathcal{J}^{k+1}(f)
+\\
+\mathcal{K}^k(f) &= \mathcal{K}^{k+1}(f)
+\end{align*}
+Ausserdem ist $l\le n$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex
new file mode 100644
index 0000000..f0bd6fa
--- /dev/null
+++ b/vorlesungen/slides/5/kernbild.tex
@@ -0,0 +1,68 @@
+%
+% kernbild.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kern und Bild}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Kern}
+Lineare Abbildung $f\colon V\to V$
+\[
+\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\}
+\]
+\end{block}
+\begin{block}{Kern von $A^k$}
+\[
+\mathcal{K}^k(f) = \operatorname{ker} f^k
+\]
+\begin{align*}
+\mathcal{K}^k(f)
+&=
+\{v\in V\;|\; f^{k}(v)=0\}
+\\
+&\subset
+\{v\in V\;|\; f^{k+1}(v)=0\}
+\\
+&=\mathcal{K}^{k+1}(f)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Bild}
+Lineare Abbildung $f\colon V\to V$
+\[
+\operatorname{im}f
+=
+\mathcal{J}(f)
+=
+\{f(v)\;|\; v\in V\}
+\]
+\end{block}
+\begin{block}{Bild von $A^k$}
+\[
+\mathcal{J}^k(f) = \operatorname{im}f^k
+\]
+\begin{align*}
+\mathcal{J}^k(f)
+&=
+\operatorname{im}f^k
+=
+\operatorname{im}(f^{k}\circ f)
+\\
+&=
+\{f^{k-1} w\;|\; w = f(v)\}
+\\
+&\subset
+\{f^{k-1} w\;|\; w \in V\}
+\\
+&\mathcal{J}^{k-1}(f)
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex
new file mode 100644
index 0000000..759d964
--- /dev/null
+++ b/vorlesungen/slides/5/ketten.tex
@@ -0,0 +1,78 @@
+%
+% ketten.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ketten von Unterräumen}
+\begin{block}{Schachtelung}
+Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt:
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcccc}
+0       &=&\mathcal{K}^0(f)
+	&\subset&\mathcal{K}^1(f)
+	&\subset&\dots
+	&\subset&\mathcal{K}^k(f)
+	&\subset&\mathcal{K}^{k+1}(f)
+	&\subset&\dots
+	&\subset&\displaystyle\bigcup_{k=0}^\infty \mathcal{K}^k(f)
+	&=:&\mathcal{K}(f)
+\\[14pt]
+\Bbbk^n &=&\mathcal{J}^0(f)
+	&\supset&\mathcal{J}^1(f)
+	&\supset&\dots
+	&\supset&\mathcal{J}^{k}(f)
+	&\supset&\mathcal{J}^{k+1}(f)
+	&\supset&\dots
+	&\supset&\displaystyle\bigcap_{k=0}^\infty \mathcal{J}^k(f)
+	&=:&\mathcal{J}(f)
+\end{array}
+\]
+\end{block}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Abildung der Kerne}
+\vspace{-10pt}
+\begin{align*}
+f \mathcal{K}^k(f)
+&=
+\{f(v)\;|\; f^k(v) = 0\}
+\\
+&\subset
+\{ v\;|\; f^{k+1}(v)=0\}
+\\
+&=
+\mathcal{K}^{k+1}(f)
+\\
+\Rightarrow
+f\mathcal{K}(f)&= f\mathcal{K}(f)
+\quad\text{invariant}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Abbildung der Bild}
+\vspace{-10pt}
+\begin{align*}
+f\mathcal{J}^k(f)
+&=
+\{f(f^{k}(v))\;|\; v\in V\}
+\\
+&=
+\{f^{k+1}(v)\;|\; v\in V\}
+\\
+&=
+\mathcal{J}^{k+1}(f)
+\\
+\Rightarrow
+f\mathcal{J}(f)&= \mathcal{J}(f)
+\quad\text{invariant}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex
new file mode 100644
index 0000000..4e8142d
--- /dev/null
+++ b/vorlesungen/slides/5/motivation.tex
@@ -0,0 +1,63 @@
+%
+% movitation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Motivation}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Matrix $A$ analysieren}
+Matrix $A$ mit Minimalpolynom $m_A(X)$ vom
+Grad $s$
+\end{block}
+\begin{block}{Faktorisieren}
+Minimalpolynom faktorisieren
+\[
+m_A(X)
+=
+(X-\mu_1)(X-\mu_2)\dots(X-\mu_s)
+\]
+\end{block}
+\begin{block}{Vertauschen}
+$\sigma\in S_s$ eine Permutation von $1,\dots,s$
+ist
+\begin{align*}
+m_A(X)
+&=
+(X-\mu_{\sigma(1)})
+%(X-\mu_{\sigma(2)})
+\dots
+(X-\mu_{\sigma(s)})
+\\
+0
+&=
+(A-\mu_{\sigma(1)})
+%(A-\mu_{\sigma(2)})
+\dots
+(A-\mu_{\sigma(s)})
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Bedingung für $\mu_k$}
+Permutation wählen so dass $\mu_k$ an erster Stelle steht:
+\[
+0=(A-\mu_k) \prod_{i\ne k}(A-\mu_i) v
+\]
+für alle $v\in\Bbbk^n$.
+\end{block}
+\begin{block}{Eigenwerte}
+Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$
+gibt mit
+\[
+(A-\mu)v = 0
+\Rightarrow Av = {\color{red}\mu} v
+\]
+Eigenwertbedingung
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/nilpotent.tex b/vorlesungen/slides/5/nilpotent.tex
new file mode 100644
index 0000000..9b7ded1
--- /dev/null
+++ b/vorlesungen/slides/5/nilpotent.tex
@@ -0,0 +1,176 @@
+%
+% nilpotent.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\feld#1{
+	\fill[color=red!20] (#1,0) rectangle ({#1+1},12);
+}
+\begin{frame}[t]
+\frametitle{$\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ für nilpotente Matrizen}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.42\textwidth}
+Matrix mit dem dargestellten Verlauf von
+${\color{red}\dim\mathcal{K}^k(A)}$
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.42]
+
+\only<2->{
+	\feld{0}
+	\feld{1}
+	\feld{2}
+	\feld{3}
+}
+\only<2->{ \feld{4} }
+\only<3->{ \feld{5} }
+\only<2->{ \feld{6} }
+\only<3->{ \feld{7} }
+\only<4->{ \feld{8} }
+\only<5->{ \feld{9} }
+\only<6->{ \feld{10} }
+\only<7->{ \feld{11} }
+
+\only<1>{ \node at (6,0) [below] {$k=0$}; }
+\only<2>{ \node at (6,0) [below] {$k=1$}; }
+\only<3>{ \node at (6,0) [below] {$k=2$}; }
+\only<4>{ \node at (6,0) [below] {$k=3$}; }
+\only<5>{ \node at (6,0) [below] {$k=4$}; }
+\only<6>{ \node at (6,0) [below] {$k=5$}; }
+\only<7>{ \node at (6,0) [below] {$k=6$}; }
+
+\draw (0,0) rectangle (12,12);
+\only<1>{
+	\foreach \x in {1,...,12}{
+		\node at ({\x-0.5},{12-\x+0.5}) {$1$};
+	}
+}
+\only<2->{
+	\foreach \x in {1,...,12}{
+		\node at ({\x-0.5},{12-\x+0.5}) {$0$};
+	}
+}
+\only<2>{
+	\foreach \x in {7,...,11}{
+		\node at ({\x+0.5},{12-\x+0.5}) {$1$};
+	}
+}
+\only<3->{
+	\foreach \x in {7,...,11}{
+		\node at ({\x+0.5},{12-\x+0.5}) {$0$};
+	}
+}
+\only<3>{
+	\foreach \x in {8,...,11}{
+		\node at ({\x+0.5},{13-\x+0.5}) {$1$};
+	}
+}
+\only<4->{
+	\foreach \x in {8,...,11}{
+		\node at ({\x+0.5},{13-\x+0.5}) {$0$};
+	}
+}
+\only<4>{
+	\foreach \x in {9,...,11}{
+		\node at ({\x+0.5},{14-\x+0.5}) {$1$};
+	}
+}
+\only<5->{
+	\foreach \x in {9,...,11}{
+		\node at ({\x+0.5},{14-\x+0.5}) {$0$};
+	}
+}
+\only<5>{
+	\foreach \x in {10,...,11}{
+		\node at ({\x+0.5},{15-\x+0.5}) {$1$};
+	}
+}
+\only<6->{
+	\foreach \x in {10,...,11}{
+		\node at ({\x+0.5},{15-\x+0.5}) {$0$};
+	}
+}
+\only<6>{
+	\foreach \x in {11,...,11}{
+		\node at ({\x+0.5},{16-\x+0.5}) {$1$};
+	}
+}
+\only<7->{
+	\foreach \x in {11,...,11}{
+		\node at ({\x+0.5},{16-\x+0.5}) {$0$};
+	}
+}
+\draw[line width=0.1pt]
+	(0,11) -- (2,11) -- (2,9) -- (4,9) -- (4,6) -- (12,6);
+\draw[line width=0.1pt]
+	(1,12) -- (1,10) -- (3,10) -- (3,8) -- (6,8) -- (6,0);
+\only<2>{
+	\node at (5.5,7.5) {$1$};
+}
+\only<3->{
+	\node at (5.5,7.5) {$0$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.56\textwidth}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\pfad{
+	(0,0) -- (1,3) -- (2,4) -- (3,4.5) -- (4,5) -- (5,5.5) -- (6,6)
+}
+\fill[color=orange!20] \pfad -- (0,6) -- cycle;
+\fill[color=darkgreen!20] \pfad -- (6,0) -- cycle;
+\foreach \y in {0.5,1,...,5.75}{
+	\draw[line width=0.1pt] (0,\y) -- (6,\y);
+}
+\draw[line width=1.4pt] \pfad;
+\draw[->] (-0.1,6) -- (6.5,6); \node at (-0.1,6) [left] {$n$};
+\draw[->] (-0.1,0) -- (6.5,0); \node at (-0.1,0) [left] {$0$};
+\fill (0,0) circle[radius=0.05];
+\fill (1,3) circle[radius=0.05];
+\fill (2,4) circle[radius=0.05];
+\fill (3,4.5) circle[radius=0.05];
+\fill (4,5) circle[radius=0.05];
+\fill (5,5.5) circle[radius=0.05];
+\fill (6,6) circle[radius=0.05];
+\only<1>{
+	\fill[color=red] (0,0) circle[radius=0.08];
+}
+\only<2>{
+	\fill[color=red] (1,3) circle[radius=0.08];
+	\draw[color=red] (0,3) -- (1,3);
+	\node[color=red] at (0,3) [left] {$6$};
+}
+\only<3>{
+	\fill[color=red] (2,4) circle[radius=0.08];
+	\draw[color=red] (0,4) -- (2,4);
+	\node[color=red] at (0,4) [left] {$8$};
+}
+\only<4>{
+	\fill[color=red] (3,4.5) circle[radius=0.08];
+	\draw[color=red] (0,4.5) -- (3,4.5);
+	\node[color=red] at (0,4.5) [left] {$9$};
+}
+\only<5>{
+	\fill[color=red] (4,5.0) circle[radius=0.08];
+	\draw[color=red] (0,5.0) -- (4,5.0);
+	\node[color=red] at (0,5.0) [left] {$10$};
+}
+\only<6>{
+	\fill[color=red] (5,5.5) circle[radius=0.08];
+	\draw[color=red] (0,5.5) -- (5,5.5);
+	\node[color=red] at (0,5.5) [left] {$11$};
+}
+\only<7>{
+	\fill[color=red] (6,6.0) circle[radius=0.08];
+}
+\draw[color=white] (-0.7,0) -- (-0.7,6);
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup