From a893dbae9b4a6c90fe837712e4e72be42154d54e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 11 Mar 2021 11:46:57 +0100 Subject: add new slides --- vorlesungen/slides/5/Aiteration.tex | 59 ++++++++++++++++++++++ vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 2 + vorlesungen/slides/5/satzvongelfand.tex | 89 +++++++++++++++++++++++++++++++++ 4 files changed, 152 insertions(+) create mode 100644 vorlesungen/slides/5/Aiteration.tex create mode 100644 vorlesungen/slides/5/satzvongelfand.tex (limited to 'vorlesungen/slides/5') 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 -- cgit v1.2.1