4 files changed, 152 insertions, 0 deletions
diff --git a/vorlesungen/slides/5/Aiteration.tex b/vorlesungen/slides/5/Aiteration.tex
new file mode 100644
index 0000000..3078c55
--- /dev/null
+++ b/vorlesungen/slides/5/Aiteration.tex
@@ -0,0 +1,59 @@
+%
+% Aiteration.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Iteration von $A$}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.34\textwidth}
+\begin{block}{$\varrho(A) > 1\uncover<4->{\Rightarrow \|A^k\|\to\infty}$}
+\uncover<2->{%
+Eigenvektor $v$, $\|v\|=1$, zum Eigenwert $\lambda$ mit $|\lambda| > 1$}
+\uncover<3->{%
+\[
+\|A^kv\| = |\lambda|^k\to \infty
+\]}
+\uncover<4->{$\Rightarrow \|A\|^k\to\infty$}
+
+\end{block}
+\end{column}
+\begin{column}{0.63\textwidth}
+\begin{block}{$\varrho(A) < 1\uncover<12->{\Rightarrow \|A\|^k\to 0}$}
+\uncover<5->{%
+$A$ setzt sich zusammen aus Jordanblöcken:
+\[
+J(\lambda)^k
+=
+\renewcommand{\arraystretch}{1.2}
+\begin{pmatrix}
+\lambda^k&\binom{k}{1}\lambda^{k-1}&\binom{k}{2}\lambda^{k-2}
+	&\dots&\binom{k}{n-1}\lambda^{k-n+1}\\
+    0    &\lambda^k&\binom{k}{1}\lambda^{k-1}
+	&\dots&\binom{k}{n-2}\lambda^{k-n+2}\\
+    0    &    0    &\lambda^k&\dots  &\binom{k}{n-3}\lambda^{k-n+3}\\
+ \vdots  & \vdots  & \vdots  &\ddots &\vdots\\
+    0    &    0    &    0    &\dots  &\lambda^k
+\end{pmatrix}
+\]}
+\uncover<6->{Alle Matrixelemente konvergieren gegen $0$:}
+\[
+\uncover<7->{\binom{k}{s} \le k^s}
+\uncover<8->{\Rightarrow
+\underbrace{\binom{k}{s}}_{\text{\uncover<9->{polynomiell $\to \infty$}}}
+\underbrace{\lambda^{k-s}}_{\text{\uncover<10->{exponentiell $\to 0$}}}
+}
+\uncover<11->{\to 0}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\uncover<13->{%
+{\usebeamercolor[fg]{title}Folgerung:}
+Es gibt $m,M$ derart, dass
+$m\varrho(A)^k \le \|A^k\|  \le M \varrho(A)^k$
+}
+\end{frame}
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
index 00c8337..e635c42 100644
--- a/vorlesungen/slides/5/Makefile.inc
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -23,6 +23,8 @@ chapter5 =								\
 	../slides/5/cayleyhamilton.tex					\
 									\
 	../slides/5/spektralgelfand.tex					\
+	../slides/5/Aiteration.tex					\
+	../slides/5/satzvongelfand.tex					\
 									\
 	../slides/5/stoneweierstrass.tex				\
 	../slides/5/potenzreihenmethode.tex				\
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
index 6f3228d..fab6a28 100644
--- a/vorlesungen/slides/5/chapter.tex
+++ b/vorlesungen/slides/5/chapter.tex
@@ -20,6 +20,8 @@
 \folie{5/reellenormalform.tex}
 \folie{5/cayleyhamilton.tex}
 \folie{5/spektralgelfand.tex}
+\folie{5/Aiteration.tex}
+\folie{5/satzvongelfand.tex}
 \folie{5/stoneweierstrass.tex}
 \folie{5/potenzreihenmethode.tex}
 \folie{5/logarithmusreihe.tex}
diff --git a/vorlesungen/slides/5/satzvongelfand.tex b/vorlesungen/slides/5/satzvongelfand.tex
new file mode 100644
index 0000000..3cf8710
--- /dev/null
+++ b/vorlesungen/slides/5/satzvongelfand.tex
@@ -0,0 +1,89 @@
+%
+% satzvongelfand.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{0pt}
+\setlength{\belowdisplayskip}{0pt}
+\setbeamercolor{block body}{bg=blue!20}
+\setbeamercolor{block title}{bg=blue!20}
+\frametitle{Satz von Gelfand}
+{\usebeamercolor[fg]{title}Behauptung:} $\varrho(A)=\pi(A)$\uncover<2->{,
+$A(\varepsilon) = \displaystyle\frac{A}{\varrho(A)+\varepsilon}$}\uncover<3->{,
+$\varrho(A(\varepsilon))=\displaystyle\frac{\varrho(A)}{\varrho(A)+\varepsilon}
+\uncover<4->{=\frac{1}{1+\varepsilon/\varrho(A)}}$}
+
+\uncover<5->{%
+%{\usebeamercolor[fg]{title}Beweisidee:}
+%$\displaystyle\pi\biggl(\frac{A}{\varrho(A)+\epsilon}\biggr)
+%=
+%\frac{\pi(A)}{\varrho(A)+\epsilon}$ berechnen
+\vspace{-5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{$\varepsilon < 0$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<6->{
+\varrho(A(\varepsilon))&>1}\uncover<7->{\quad\Rightarrow\quad \|A(\varepsilon)^k\|\to \infty}
+\\
+\uncover<8->{\|A(\varepsilon)^k\| &\ge m\varrho(A(\varepsilon))^k}
+\\
+\uncover<9->{\|A(\varepsilon)^k\|^{\frac1k} &\ge m^{\frac1k} \varrho(A(\varepsilon))}
+\\
+\uncover<10->{\pi(A) &\ge \lim_{k\to\infty}m^{\frac1k}\varrho(A(\varepsilon))}
+\\
+&\uncover<11->{= \varrho(A(\varepsilon))}\uncover<12->{ > 1}
+\\
+\uncover<13->{\frac{ \pi(A(\varepsilon))}{\varrho(A)+\varepsilon} &> 1}
+\\
+\uncover<14->{
+\pi(A) &> \varrho(A)+\varepsilon
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\varepsilon > 0$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<16->{
+\varrho(A(\varepsilon)) &<1}
+\uncover<17->{\quad\Rightarrow\quad \|A(\varepsilon)^k\| \to 0}
+\\
+\uncover<18->{\|A(\varepsilon)^k\|
+&\le M\varrho(A(\varepsilon))^k}
+\\
+\uncover<19->{
+\|A(\varepsilon)^k\|^{\frac1k}
+&\le M^{\frac1k}\varrho(A(\varepsilon))
+}
+\\
+\uncover<20->{
+\pi(A(\varepsilon))
+&\le
+\varrho(A(\varepsilon)) \lim_{k\to\infty} M^{\frac1k}
+}
+\\
+&\uncover<21->{= \varrho(A(\varepsilon))}
+\uncover<22->{ < 1}
+\\
+\uncover<23->{\frac{\pi(A)}{\varrho(A)+\varepsilon}&< 1}
+\\
+\uncover<24->{\pi(A)&< \varrho(A) + \varepsilon}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}}
+\uncover<15->{%
+\vspace{2pt}
+{\usebeamercolor[fg]{title}Folgerung:}
+$\varrho(A)-\varepsilon < \pi(A) \uncover<25->{< \varrho(A)+\varepsilon}\quad\forall\varepsilon>0
+\uncover<26->{
+\qquad\Rightarrow\qquad
+\varrho(A)=\pi(A)}$
+}
+\end{frame}
+\egroup