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-rw-r--r--vorlesungen/02_msespektral/slides.tex11
-rw-r--r--vorlesungen/slides/5/Makefile.inc3
-rw-r--r--vorlesungen/slides/5/bloecke.tex141
-rw-r--r--vorlesungen/slides/5/chapter.tex3
-rw-r--r--vorlesungen/slides/5/charpoly.tex26
-rw-r--r--vorlesungen/slides/5/eigenraeume.tex14
-rw-r--r--vorlesungen/slides/5/folgerungen.tex62
-rw-r--r--vorlesungen/slides/5/injektiv.tex74
-rw-r--r--vorlesungen/slides/5/jordan.tex130
-rw-r--r--vorlesungen/slides/5/kernbild.tex28
-rw-r--r--vorlesungen/slides/5/ketten.tex3
-rw-r--r--vorlesungen/slides/5/motivation.tex12
-rw-r--r--vorlesungen/slides/5/normalnilp.tex229
-rw-r--r--vorlesungen/slides/5/zerlegung.tex44
-rw-r--r--vorlesungen/slides/test.tex28
15 files changed, 756 insertions, 52 deletions
diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex
index dc34236..9be6ce1 100644
--- a/vorlesungen/02_msespektral/slides.tex
+++ b/vorlesungen/02_msespektral/slides.tex
@@ -18,18 +18,23 @@
\folie{5/ketten.tex}
\folie{5/dimension.tex}
\folie{5/folgerungen.tex}
+\folie{5/injektiv.tex}
\folie{5/nilpotent.tex}
-% XXX \folie{5/eigenraeume.tex}
+\folie{5/eigenraeume.tex}
+\folie{5/zerlegung.tex}
+\folie{5/normalnilp.tex}
+\folie{5/bloecke.tex}
% Jordan Normalform
\section{Jordan-Normalform}
% XXX Diagonalform
% XXX \folie{5/diagonalform.tex}
-% XXX \folie{5/jordannormalform.tex}
+\folie{5/jordanblock.tex}
+\folie{5/jordan.tex}
% XXX \folie{5/minimalpolynom.tex}
% XXX \folie{5/reellenormalform.tex}
% XXX \folie{5/hessenberg.tex}
\section{Satz von Cayley-Hamilton}
-% XXX \folie{5/cayleyhamilton.tex}
+\folie{5/cayleyhamilton.tex}
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
index 4bed49b..872798e 100644
--- a/vorlesungen/slides/5/Makefile.inc
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -16,6 +16,9 @@ chapter5 = \
../slides/5/eigenraeume.tex \
../slides/5/zerlegung.tex \
../slides/5/normalnilp.tex \
+ ../slides/5/bloecke.tex \
+ ../slides/5/jordanblock.tex \
../slides/5/jordan.tex \
+ ../slides/5/cayleyhamilton.tex \
../slides/5/chapter.tex
diff --git a/vorlesungen/slides/5/bloecke.tex b/vorlesungen/slides/5/bloecke.tex
new file mode 100644
index 0000000..974f238
--- /dev/null
+++ b/vorlesungen/slides/5/bloecke.tex
@@ -0,0 +1,141 @@
+%
+% bloecke.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1}
+\def\sy{0.1}
+\def\block#1#2{
+ \fill[color=red] ({#1},{-#1}) rectangle ({#1+#2},{-#1-#2});
+}
+\def\kreuz#1{
+ \draw[color=white,line width=0.1pt] (0,{-#1})--(60,{-#1});
+ \draw[color=white,line width=0.1pt] (#1,0)--(#1,-60);
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Blockgrössen aus $\dim\mathcal{K}^k(A)$ ablesen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.56\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\coordinate (A) at ({1*\sx},{20*\sy});
+\coordinate (B) at ({2*\sx},{(20+15)*\sy});
+\coordinate (C) at ({3*\sx},{(20+15+10)*\sy});
+\coordinate (D) at ({4*\sx},{(20+15+10+8)*\sy});
+\coordinate (E) at ({5*\sx},{(20+15+10+8+5)*\sy});
+\coordinate (F) at ({6*\sx},{(20+15+10+8+5+2)*\sy});
+\fill[color=darkgreen!20] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F)
+ -- ({6*\sx},0) -- cycle;
+
+\fill[color=darkgreen!40] (0,0) -- ({1*\sx},0) -- (A) -- cycle;
+\fill[color=darkgreen!40] (A) -- ({2*\sx},{20*\sy}) -- (B) -- cycle;
+\fill[color=darkgreen!40] (B) -- ({3*\sx},{(20+15)*\sy}) -- (C) -- cycle;
+\fill[color=darkgreen!40] (C) -- ({4*\sx},{(20+15+10)*\sy}) -- (D) -- cycle;
+\fill[color=darkgreen!40] (D) -- ({5*\sx},{(20+15+10+8)*\sy}) -- (E) -- cycle;
+\fill[color=darkgreen!40] (E) -- ({6*\sx},{(20+15+10+8+5)*\sy}) -- (F) -- cycle;
+
+\draw[color=darkgreen,line width=1.4pt] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F);
+
+\draw[color=gray] (A) -- (0,{20*\sy});
+\draw[color=gray] (B) -- (0,{(20+15)*\sy});
+\draw[color=gray] (C) -- (0,{(20+15+10)*\sy});
+\draw[color=gray] (D) -- (0,{(20+15+10+8)*\sy});
+\draw[color=gray] (E) -- (0,{(20+15+10+8+5)*\sy});
+\draw[color=gray] (F) -- (0,{(20+15+10+8+5+2)*\sy});
+
+\node at ({0.5*\sx},{0.5*20*\sy})
+ [right] {$d_1 = \dim\mathcal{K}^1(A)-\dim\mathcal{K}^0(A)$};
+\node at ({1.5*\sx},{0.5*(20+20+15)*\sy})
+ [right] {$d_2 = \dim\mathcal{K}^2(A)-\dim\mathcal{K}^1(A)$};
+\node at ({2.5*\sx},{0.5*(2*20+2*15+1*10)*\sy}) [right] {$d_3$};
+\node at ({3.5*\sx},{0.5*(2*20+2*15+2*10+8)*\sy}) [right] {$d_4$};
+\node at ({4.5*\sx-0.1},{0.5*(2*20+2*15+2*10+2*8+5)*\sy+0.2}) [below right] {$d_5$};
+\node at ({5.5*\sx},{0.5*(2*20+2*15+2*10+2*8+2*5+2)*\sy+0.1}) [below] {$d_6$};
+
+\fill (A) circle[radius=0.08];
+\fill (B) circle[radius=0.08];
+\fill (C) circle[radius=0.08];
+\fill (D) circle[radius=0.08];
+\fill (E) circle[radius=0.08];
+\fill (F) circle[radius=0.08];
+
+\draw[->] (-0.1,0) -- ({6*\sx+1},0) coordinate[label={$k$}];
+\draw[->] (0,-0.1) -- (0,6.5) coordinate[label={right:$\dim\mathcal{K}^k(A)$}];
+
+\foreach \x in {0,1,...,6}{
+ \draw ({\sx*\x},{-0.05}) -- ({\sx*\x},0.05);
+ \node at ({\sx*\x},{-0.1}) [below] {$\x$};
+}
+
+\node at (0,{60*\sy}) [left] {\llap{$n$}};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.43\textwidth}
+\vspace{-10pt}
+\begin{center}
+\begin{tabular}{>{$}c<{$}|>{$}r<{$}|>{$}c<{$}|>{$}c<{$}}
+k&d_k&\# M_k(\Bbbk)\text{-Blöcke}&\text{Beispiel}\\
+\hline
+0& 0& &\\
+1& 20& d_1-d_2&5\\
+2& 15& d_2-d_3&5\\
+3& 10& d_3-d_4&2\\
+4& 8& d_4-d_5&3\\
+5& 5& d_5-d_6&3\\
+6& 2& d_6 &2\\
+\end{tabular}
+\end{center}
+\vspace{-13pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.05]
+\fill[color=gray!40] (0,0) rectangle (60,-60);
+\node[color=white] at (30,-30) [scale=6] {$A$};
+\kreuz{5}
+\kreuz{15}
+\kreuz{21}
+\kreuz{33}
+\kreuz{48}
+\node at (0,-2.5) [left] {$k=1$};
+\node at (60,-2.5) [right] {$5$ Blöcke};
+\node at (0,-10) [left] {$k=2$};
+\node at (60,-10) [right] {$5$ Blöcke};
+\node at (0,-18) [left] {$k=3$};
+\node at (60,-18) [right] {$2$ Blöcke};
+\node at (0,-27) [left] {$k=4$};
+\node at (60,-27) [right] {$3$ Blöcke};
+\node at (0,-40.5) [left] {$k=5$};
+\node at (60,-40.5) [right] {$3$ Blöcke};
+\node at (0,-54) [left] {$k=6$};
+\node at (60,-54) [right] {$2$ Blöcke};
+\block{0}{1}
+\block{1}{1}
+\block{2}{1}
+\block{3}{1}
+\block{4}{1}
+\block{5}{2}
+\block{7}{2}
+\block{9}{2}
+\block{11}{2}
+\block{13}{2}
+\block{15}{3}
+\block{18}{3}
+\block{21}{4}
+\block{25}{4}
+\block{29}{4}
+\block{33}{5}
+\block{38}{5}
+\block{43}{5}
+\block{48}{6}
+\block{54}{6}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
index 5083abf..7698f01 100644
--- a/vorlesungen/slides/5/chapter.tex
+++ b/vorlesungen/slides/5/chapter.tex
@@ -14,4 +14,7 @@ folie{5/nilpotent.tex}
folie{5/eigenraeume.tex}
folie{5/zerlegung.tex}
folie{5/normalnilp.tex}
+folie{5/bloecke.tex}
+folie{5/jordanblock.tex}
folie{5/jordan.tex}
+folie{5/cayleyhamilton.tex}
diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex
index 1211b43..63bfee5 100644
--- a/vorlesungen/slides/5/charpoly.tex
+++ b/vorlesungen/slides/5/charpoly.tex
@@ -20,6 +20,7 @@ $A-\mu I$ singulär ist:
\]
$\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$
\end{block}
+\uncover<2->{%
\begin{block}{Zerlegung in Linearfaktoren}
$\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$:
\[
@@ -27,33 +28,42 @@ $\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$:
=
(X-\mu_1)\dots (X-\mu_n)
\]
-\end{block}
+\end{block}}
+\uncover<3->{%
\begin{block}{Fundamentalsatz der Algebra}
Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in
Linearfaktoren
-\end{block}
+\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
+\uncover<4->{%
\begin{block}{Minimalpolynom}
Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen
-\end{block}
+\end{block}}
+\uncover<5->{%
\begin{proof}[Beweis]
\begin{enumerate}
-\item
+\item<6->
$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$
-\item
+\item<7->
$A-\lambda I$ ist regulär
\end{enumerate}
+\uncover<8->{%
\begin{align*}
&\Rightarrow&
m_A(A)&=0
\\
&&
+\uncover<9->{
(A-\lambda)^{-1}m_A(A) &=0
+}
\\
&&
+\uncover<10->{
\prod_{i\in I}(A-\mu_i)&=0,
-\end{align*}
+}
+\end{align*}}
+\uncover<11->{%
d.~h.~\(
\displaystyle
\overline{m}_A(X)
@@ -61,8 +71,8 @@ d.~h.~\(
\prod_i{i\in I}(X-\mu_i)
\in
\mathbb{C}[X]
-\)
-\end{proof}
+\)}
+\end{proof}}
\end{column}
\end{columns}
\end{frame}
diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex
index 5192cbc..fd4803c 100644
--- a/vorlesungen/slides/5/eigenraeume.tex
+++ b/vorlesungen/slides/5/eigenraeume.tex
@@ -15,19 +15,25 @@ E_\lambda(f)
&=
\ker (f-\lambda)
\\
+\uncover<2->{
&=
\{v\in V\;|\; f(v) = \lambda v\}
+}
\end{align*}
-{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.
+\uncover<3->{%
+{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.}
\end{block}
-$E_\lambda(f)\subset V$ ist ein Unterraum
+\uncover<4->{%
+$E_\lambda(f)\subset V$ ist ein Unterraum}
+\uncover<5->{%
\begin{block}{Eigenwert}
Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$.
-\end{block}
+\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
+\uncover<6->{%
\begin{block}{verallgemeinerter Eigenraum}
Für $\lambda\in \Bbbk$ heisst
\[
@@ -36,7 +42,7 @@ Für $\lambda\in \Bbbk$ heisst
\mathcal{K}(f-\lambda)
\]
verallgemeinerter Eigenraum
-\end{block}
+\end{block}}
\end{column}
\end{columns}
\end{frame}
diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex
index a1fa6bf..4a8dbe6 100644
--- a/vorlesungen/slides/5/folgerungen.tex
+++ b/vorlesungen/slides/5/folgerungen.tex
@@ -3,10 +3,14 @@
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
+\bgroup
+\def\sx{1}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{frame}[t]
\frametitle{Folgerungen}
+\vspace{-10pt}
\begin{columns}[t]
-\begin{column}{0.48\textwidth}
+\begin{column}{0.30\textwidth}
\begin{block}{Zunahme}
Für alle $k<l$ gilt
\begin{align*}
@@ -23,10 +27,58 @@ Für $k\ge l$ gilt
Ausserdem ist $l\le n$
\end{block}
\end{column}
-\begin{column}{0.48\textwidth}
-\begin{proof}[Beweis]
-XXX
-\end{proof}
+\begin{column}{0.66\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\pfad{
+ ({0*\sx},6) --
+ ({1*\sx},4.5) --
+ ({2*\sx},3.5) --
+ ({3*\sx},2.9) --
+ ({4*\sx},2.6) --
+ ({5*\sx},2.4) --
+ ({6*\sx},2.4)
+}
+
+\fill[color=orange!20] \pfad -- ({6*\sx},0) -- (0,0) -- cycle;
+\fill[color=darkgreen!20] \pfad -- ({6*\sx},6) -- cycle;
+\fill[color=orange!40] ({5*\sx},0) rectangle ({6*\sx},2.4);
+\fill[color=darkgreen!40] ({5*\sx},6) rectangle ({6*\sx},2.4);
+
+\draw[color=darkgreen,line width=2pt] ({3*\sx},6) -- ({3*\sx},2.9);
+\node[color=darkgreen] at ({3*\sx},4.45) [rotate=90,above] {$\dim\mathcal{K}^k(A)$};
+\draw[color=orange,line width=2pt] ({3*\sx},0) -- ({3*\sx},2.9);
+\node[color=orange] at ({3*\sx},1.45) [rotate=90,above] {$\dim\mathcal{J}^k(A)$};
+
+\node[color=orange] at ({5.5*\sx},1.2) [rotate=90] {bijektiv};
+\node[color=darkgreen] at ({5.5*\sx},4.2) [rotate=90] {konstant};
+
+\fill ({0*\sx},6) circle[radius=0.08];
+\fill ({1*\sx},4.5) circle[radius=0.08];
+\fill ({2*\sx},3.5) circle[radius=0.08];
+\fill ({3*\sx},2.9) circle[radius=0.08];
+\fill ({4*\sx},2.6) circle[radius=0.08];
+\fill ({5*\sx},2.4) circle[radius=0.08];
+\fill ({6*\sx},2.4) circle[radius=0.08];
+
+\draw \pfad;
+
+\draw[->] (-0.1,0) -- ({6*\sx+0.5},0) coordinate[label={$k$}];
+\draw[->] (-0.1,6) -- ({6*\sx+0.5},6);
+
+\foreach \x in {0,...,6}{
+ \draw (\x,-0.05) -- (\x,0.05);
+}
+\foreach \x in {0,...,3}{
+ \node at ({\x*\sx},-0.05) [below] {$\x$};
+}
+\node at ({4*\sx},-0.05) [below] {$\dots\mathstrut$};
+\node at ({5*\sx},-0.05) [below] {$l$};
+\node at ({6*\sx},-0.05) [below] {$l+1$};
+
+\end{tikzpicture}
+\end{center}
\end{column}
\end{columns}
\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex
index e08cafd..90cbcd6 100644
--- a/vorlesungen/slides/5/injektiv.tex
+++ b/vorlesungen/slides/5/injektiv.tex
@@ -3,7 +3,79 @@
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
+\bgroup
+\def\sx{1.05}
\begin{frame}[t]
\frametitle{$f$ injektiv auf $\mathcal{J}(f)$}
-XXX
+\setlength{\abovedisplayskip}{8pt}
+\setlength{\belowdisplayskip}{8pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=orange!20]
+ ({0*\sx},-3.0) -- ({1*\sx},-2.0) -- ({2*\sx},-1.5) --
+ ({3*\sx},-1.1) -- ({4*\sx},-0.9) -- ({5*\sx},-0.8) --
+ ({6*\sx},-0.8) --
+ ({6*\sx},0.8) -- ({5*\sx},0.8) -- ({4*\sx},0.9) --
+ ({3*\sx},1.1) -- ({2*\sx},1.5) -- ({1*\sx},2.0) --
+ ({0*\sx},3.0) -- cycle;
+\fill[color=orange!40] (0,-0.8) rectangle ({6*\sx},0.8);
+
+\foreach \x in {0,...,6}{
+ \draw[color=gray,line width=3pt] ({\x*\sx},-3)--({\sx*\x},3);
+}
+\foreach \x in {0,1,2,3}{
+ \node at ({\sx*\x},-3) [below] {$\x$};
+}
+\node at ({\sx*5},-3) [below] {$l$};
+\node at ({\sx*6},-3) [below] {$l+1$};
+\draw[->] (-0.1,-3.5) -- ({6*\sx+0.4},-3.5) coordinate[label={below:$k$}];
+
+\draw[line width=3pt,color=orange] ({0*\sx},-3.0) -- ({0*\sx},3.0);
+\draw[line width=3pt,color=orange] ({1*\sx},-2.0) -- ({1*\sx},2.0);
+\draw[line width=3pt,color=orange] ({2*\sx},-1.5) -- ({2*\sx},1.5);
+\draw[line width=3pt,color=orange] ({3*\sx},-1.1) -- ({3*\sx},1.1);
+\draw[line width=3pt,color=orange] ({4*\sx},-0.9) -- ({4*\sx},0.9);
+\draw[line width=3pt,color=orange] ({5*\sx},-0.8) -- ({5*\sx},0.8);
+\draw[line width=3pt,color=orange] ({6*\sx},-0.8) -- ({6*\sx},0.8);
+
+\foreach \x in {0,1,2,3}{
+ \node at ({\x*\sx},0) [rotate=90] {$\mathcal{J}^{\x}(A)$};
+}
+\node at ({4*\sx},0) {$\cdots$};
+\node at ({5*\sx},0) [rotate=90] {$\mathcal{J}^{l}(A)$};
+\node at ({6*\sx},0) [rotate=90] {$\mathcal{J}^{l+1}(A)$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.38\textwidth}
+\begin{block}{stationär}
+$l$ der $k$-Wert, ab dem gilt
+\begin{align*}
+\mathcal{J}^l(A) &= \mathcal{J}^{l+1}(A) = A\mathcal{J}^l(A)
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Dimension}
+\vspace{-10pt}
+\[
+\dim \mathcal{J}^l(A) = \dim\mathcal{J}^{l+1}(A)
+\]
+\uncover<3->{%
+d.~h.~$A$ ist bijektiv als Selbstabbildung von
+$\mathcal{J}(A)$}
+\uncover<4->{%
+\[
+\Downarrow
+\]
+$A|\mathcal{J}(A)$ ist injektiv}
+\end{block}}
+\end{column}
+\end{columns}
\end{frame}
+\egroup
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
diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex
index f0bd6fa..3890717 100644
--- a/vorlesungen/slides/5/kernbild.tex
+++ b/vorlesungen/slides/5/kernbild.tex
@@ -10,29 +10,38 @@
\vspace{-15pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
+\uncover<1->{%
\begin{block}{Kern}
Lineare Abbildung $f\colon V\to V$
\[
\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\}
\]
-\end{block}
+\end{block}}
+\uncover<3->{%
\begin{block}{Kern von $A^k$}
\[
\mathcal{K}^k(f) = \operatorname{ker} f^k
\]
\begin{align*}
+\uncover<5->{
\mathcal{K}^k(f)
&=
\{v\in V\;|\; f^{k}(v)=0\}
+}
\\
+\uncover<6->{
&\subset
\{v\in V\;|\; f^{k+1}(v)=0\}
+}
\\
+\uncover<7->{
&=\mathcal{K}^{k+1}(f)
+}
\end{align*}
-\end{block}
+\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
+\uncover<2->{%
\begin{block}{Bild}
Lineare Abbildung $f\colon V\to V$
\[
@@ -42,27 +51,36 @@ Lineare Abbildung $f\colon V\to V$
=
\{f(v)\;|\; v\in V\}
\]
-\end{block}
+\end{block}}
+\uncover<4->{%
\begin{block}{Bild von $A^k$}
\[
\mathcal{J}^k(f) = \operatorname{im}f^k
\]
\begin{align*}
+\uncover<8->{
\mathcal{J}^k(f)
&=
\operatorname{im}f^k
=
\operatorname{im}(f^{k}\circ f)
+}
\\
+\uncover<9->{
&=
\{f^{k-1} w\;|\; w = f(v)\}
+}
\\
+\uncover<10->{
&\subset
\{f^{k-1} w\;|\; w \in V\}
+}
\\
-&\mathcal{J}^{k-1}(f)
+\uncover<11->{
+&=\mathcal{J}^{k-1}(f)
+}
\end{align*}
-\end{block}
+\end{block}}
\end{column}
\end{columns}
\end{frame}
diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex
index 759d964..1116a83 100644
--- a/vorlesungen/slides/5/ketten.tex
+++ b/vorlesungen/slides/5/ketten.tex
@@ -33,6 +33,7 @@ Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt:
\]
\end{block}
\vspace{-20pt}
+\uncover<2->{%
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Abildung der Kerne}
@@ -74,5 +75,5 @@ f\mathcal{J}(f)&= \mathcal{J}(f)
\end{align*}
\end{block}
\end{column}
-\end{columns}
+\end{columns}}
\end{frame}
diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex
index 4e8142d..b0a1d82 100644
--- a/vorlesungen/slides/5/motivation.tex
+++ b/vorlesungen/slides/5/motivation.tex
@@ -13,6 +13,7 @@
Matrix $A$ mit Minimalpolynom $m_A(X)$ vom
Grad $s$
\end{block}
+\uncover<2->{%
\begin{block}{Faktorisieren}
Minimalpolynom faktorisieren
\[
@@ -20,7 +21,8 @@ m_A(X)
=
(X-\mu_1)(X-\mu_2)\dots(X-\mu_s)
\]
-\end{block}
+\end{block}}
+\uncover<3->{%
\begin{block}{Vertauschen}
$\sigma\in S_s$ eine Permutation von $1,\dots,s$
ist
@@ -39,16 +41,18 @@ m_A(X)
\dots
(A-\mu_{\sigma(s)})
\end{align*}
-\end{block}
+\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
+\uncover<4->{%
\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}
+\end{block}}
+\uncover<5->{%
\begin{block}{Eigenwerte}
Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$
gibt mit
@@ -57,7 +61,7 @@ gibt mit
\Rightarrow Av = {\color{red}\mu} v
\]
Eigenwertbedingung
-\end{block}
+\end{block}}
\end{column}
\end{columns}
\end{frame}
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
diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex
index 9c20c60..a734d69 100644
--- a/vorlesungen/slides/5/zerlegung.tex
+++ b/vorlesungen/slides/5/zerlegung.tex
@@ -14,27 +14,48 @@
\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] (0,6) -- (15,6);
-\draw[line width=0.1pt] (0,3) -- (15,3);
\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 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) {$f_{|\mathcal{E}_{\lambda_1}}$};
+}
+\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}
@@ -42,30 +63,41 @@
\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{align*}}
\end{block}
\end{column}
\end{columns}
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index c2d361f..2538f29 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -48,31 +48,31 @@
\section{Eigenwertproblem}
% XXX Motivation: beliebige Funktionen f(A) berechnen
-\folie{5/motivation.tex}
-\folie{5/charpoly.tex}
+%\folie{5/motivation.tex}
+%\folie{5/charpoly.tex}
\section{Invariante Unterräume}
-\folie{5/kernbild.tex}
-\folie{5/ketten.tex}
-\folie{5/dimension.tex}
-\folie{5/folgerungen.tex}
-\folie{5/injektiv.tex}
-\folie{5/nilpotent.tex}
-\folie{5/eigenraeume.tex}
-\folie{5/zerlegung.tex}
-\folie{5/normalnilp.tex}
+%\folie{5/kernbild.tex}
+%\folie{5/ketten.tex}
+%\folie{5/dimension.tex}
+%\folie{5/folgerungen.tex}
+%\folie{5/injektiv.tex}
+%\folie{5/nilpotent.tex}
+%\folie{5/eigenraeume.tex}
+%\folie{5/zerlegung.tex}
+%\folie{5/normalnilp.tex}
+%\folie{5/bloecke.tex}
% Jordan Normalform
\section{Jordan-Normalform}
+\folie{5/jordanblock.tex}
\folie{5/jordan.tex}
% XXX Diagonalform
% XXX \folie{5/diagonalform.tex}
-% XXX \folie{5/jordannormalform.tex}
-% XXX \folie{5/minimalpolynom.tex}
% XXX \folie{5/reellenormalform.tex}
% XXX \folie{5/hessenberg.tex}
\section{Satz von Cayley-Hamilton}
-% XXX \folie{5/cayleyhamilton.tex}
+\folie{5/cayleyhamilton.tex}
generated by cgit v1.2.1 (git 2.18.0) at 2024-06-16 20:55:47 +0200