add new slides

7 files changed, 143 insertions, 2 deletions

diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
index d081b29..0975571 100644
--- a/vorlesungen/slides/5/Makefile.inc
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -11,6 +11,10 @@ chapter5 =								\
 	../slides/5/ketten.tex						\
 	../slides/5/dimension.tex					\
 	../slides/5/folgerungen.tex					\
+	../slides/5/injektiv.tex					\
 	../slides/5/nilpotent.tex					\
+	../slides/5/eigenraeume.tex					\
+	../slides/5/zerlegung.tex					\
+	../slides/5/normalnilp.tex					\
 	../slides/5/chapter.tex
 
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
index 4bcee8e..245b31d 100644
--- a/vorlesungen/slides/5/chapter.tex
+++ b/vorlesungen/slides/5/chapter.tex
@@ -9,4 +9,8 @@ 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}
diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex
new file mode 100644
index 0000000..5192cbc
--- /dev/null
+++ b/vorlesungen/slides/5/eigenraeume.tex
@@ -0,0 +1,42 @@
+%
+% eigenraeume.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Eigenräume}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenraum}
+Für $\lambda\in\Bbbk$ heisst
+\begin{align*}
+E_\lambda(f)
+&=
+\ker (f-\lambda)
+\\
+&=
+\{v\in V\;|\; f(v) = \lambda v\}
+\end{align*}
+{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.
+\end{block}
+$E_\lambda(f)\subset V$ ist ein Unterraum
+
+\begin{block}{Eigenwert}
+Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$.
+\end{block}
+
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{verallgemeinerter Eigenraum}
+Für $\lambda\in \Bbbk$ heisst
+\[
+\mathcal{E}_\lambda(f)
+=
+\mathcal{K}(f-\lambda)
+\]
+verallgemeinerter Eigenraum
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex
index 96efd7f..a1fa6bf 100644
--- a/vorlesungen/slides/5/folgerungen.tex
+++ b/vorlesungen/slides/5/folgerungen.tex
@@ -24,8 +24,9 @@ Ausserdem ist $l\le n$
 \end{block}
 \end{column}
 \begin{column}{0.48\textwidth}
-\begin{block}{}
-\end{block}
+\begin{proof}[Beweis]
+XXX
+\end{proof}
 \end{column}
 \end{columns}
 \end{frame}
diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex
new file mode 100644
index 0000000..e08cafd
--- /dev/null
+++ b/vorlesungen/slides/5/injektiv.tex
@@ -0,0 +1,9 @@
+%
+% injektiv.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{$f$ injektiv auf $\mathcal{J}(f)$}
+XXX
+\end{frame}
diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex
new file mode 100644
index 0000000..a7af682
--- /dev/null
+++ b/vorlesungen/slides/5/normalnilp.tex
@@ -0,0 +1,8 @@
+%
+% normalnilp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Normalform einer nilpotenten Matrix}
+\end{frame}
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