%
% spektralgelfand.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\def\eigenwert#1#2{
	\fill[color=blue!30] (#1:#2) circle[radius=0.05];
	\draw[color=blue] (#1:#2) circle[radius=0.05];
}
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Spektral- und Gelfand-Radius}
\vspace{-15pt}
\begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth}
\begin{block}{Spektralradius}
\vspace{-10pt}
\[
\varrho(A)
=
\sup\{|\lambda|\;|\; \text{{\color{blue}$\lambda$} ist EW von $A$}\}
\]
\begin{center}
\begin{tikzpicture}[>=latex,thick]
\uncover<5->{
	\fill[color=red!30] (0,0) circle[radius=2.2];
	\draw[color=red] (0,0) circle[radius=2.2];
}

\uncover<3->{
	\eigenwert{190.46}{1.3365}
	%\eigenwert{52.663}{2.1819}
	\eigenwert{281.94}{1.7305}
	\eigenwert{21.29}{1.0406}
	\eigenwert{69.511}{1.56}
	\eigenwert{63.365}{1.3535}
	\eigenwert{281.43}{0.31994}
	\eigenwert{313.1}{1.5419}
	\eigenwert{118.14}{1.1966}
	\eigenwert{195.75}{0.41156}
	\eigenwert{233.42}{1.5613}
	\eigenwert{25.203}{1.1936}
	\eigenwert{53.375}{1.4886}
	\eigenwert{346.13}{2.1073}
	\eigenwert{246.47}{1.124}
	\eigenwert{35.451}{1.99}
	\eigenwert{212.43}{1.9708}
	\eigenwert{58.479}{0.61602}
	\eigenwert{344.37}{1.6107}
	\eigenwert{305.42}{1.7075}
	\eigenwert{29.693}{0.28791}
	\eigenwert{195.82}{0.63079}
	\eigenwert{209.71}{0.25669}
	\eigenwert{51.355}{0.7247}
	\eigenwert{356.43}{1.0867}
	\eigenwert{33.119}{0.7328}
	\eigenwert{73.131}{1.5021}
	\eigenwert{345.67}{0.37564}
	\eigenwert{76.52}{0.71763}
	%\eigenwert{197.04}{2.1431}
	\eigenwert{217.87}{1.7704}
	\eigenwert{172.93}{1.1204}
	\eigenwert{339.19}{1.5305}
	\eigenwert{272.86}{2.04}
	\eigenwert{168.8}{1.6289}
	\eigenwert{248.68}{0.70879}
	\eigenwert{237.98}{0.71097}
	\eigenwert{81.411}{1.8461}
	\eigenwert{224.65}{1.0827}
	\eigenwert{357.54}{0.291}
	\eigenwert{325.26}{1.2778}
	\eigenwert{150.97}{0.32358}
	\eigenwert{260.68}{1.4077}
	\eigenwert{116.29}{1.0715}
	\eigenwert{358.25}{0.99667}
	\eigenwert{276.2}{0.077375}
	\eigenwert{316.16}{0.77763}
	\eigenwert{69.398}{1.2818}
	\eigenwert{353.5}{0.74099}
	\eigenwert{4.7935}{1.391}
	\eigenwert{136.98}{1.7572}
	\eigenwert{45.62}{1.9649}
	\eigenwert{299.96}{0.19199}
	\eigenwert{187.32}{0.63805}
	\eigenwert{272.88}{1.1467}
	\eigenwert{231.85}{1.5763}
	\eigenwert{124.24}{0.77024}
	\eigenwert{196.24}{2.0375}
	\eigenwert{186.33}{1.0656}
	%\eigenwert{22.812}{2.1616}
	\eigenwert{37.982}{0.038956}
	\eigenwert{142.36}{1.7944}
	\eigenwert{56.863}{1.8952}
	\eigenwert{4.6281}{1.1857}
	\eigenwert{71.674}{0.07642}
	\eigenwert{94.049}{1.8985}
	\eigenwert{97.294}{0.23412}
	\eigenwert{84.739}{0.31209}
	\eigenwert{147.42}{1.8434}
	\eigenwert{160.67}{0.76956}
	\eigenwert{292.5}{0.85697}
	\eigenwert{308.1}{1.7061}
	\eigenwert{68.669}{2.111}
	\eigenwert{86.866}{1.1271}
	\eigenwert{124.72}{1.3019}
	\eigenwert{267.36}{0.7462}
	\eigenwert{295.78}{1.0425}
	\eigenwert{44.972}{0.65363}
	\eigenwert{34.534}{1.2817}
	\eigenwert{357.78}{2.0592}
	\eigenwert{147.52}{0.020535}
	%\eigenwert{28.502}{2.1964}
	\eigenwert{343.48}{2.0968}
	\eigenwert{129.96}{0.80371}
	\eigenwert{254.75}{1.5775}
	\eigenwert{89.91}{0.88605}
	\eigenwert{20.35}{0.66065}
	\eigenwert{60.382}{1.7585}
	\eigenwert{158.87}{0.68399}
	\eigenwert{328.44}{1.504}
	\eigenwert{189.41}{0.33879}
	\eigenwert{273.47}{0.11109}
	\eigenwert{285.99}{0.66704}
	\eigenwert{311.42}{2.0266}
	\eigenwert{32.636}{0.5713}
	\eigenwert{221.35}{2.1329}
	\eigenwert{50.983}{1.1957}
	\eigenwert{53.298}{1.2982}
	\eigenwert{101.4}{1.9051}
	\eigenwert{71.999}{0.25671}
}

\uncover<2->{
	\draw[->] (-2.4,0) -- (2.7,0)
		coordinate[label={$\operatorname{Re}z$}];
	\draw[->] (0,-2.4) -- (0,2.5)
		coordinate[label={right:$\operatorname{Im}z$}];
}
\uncover<4->{
	\fill[color=darkgreen] (0,0) circle[radius=0.05];
	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (150:2.2);
	\node[color=darkgreen] at ($(150:1.85)+(0.4,0)$)
		[below left] {$\varrho(A)$};
}
\uncover<3->{
	\eigenwert{150}{2.2}
}
\end{tikzpicture}
\end{center}
\end{block}
\end{column}
\begin{column}{0.48\textwidth}
\uncover<6->{%
\begin{block}{Gelfand-Radius}
\[
\pi(A)
=
\lim_{k\to\infty} \|A^k\|^{\frac{1}{k}}
\]
\end{block}}
\vspace{-8pt}
\uncover<7->{%
\begin{block}{Konvergenz der Neumann-Reihe}
$
\uncover<8->{t<1/\pi(A)\;}
\uncover<10->{\Rightarrow\; \exists q}
\uncover<11->{,N}$
\begin{align*}
\uncover<9->{ t\pi(A) & \only<10->{< q} < 1 }
\\
\uncover<11->{ \|(tA)^k\|^{\frac1k} &\le q }
\\
\uncover<12->{
\|(tA)^k\|
&\le
(t\pi(A))^k<q^k
}
\end{align*}
\uncover<11->{für $k>N$.}
\uncover<13->{
$\Rightarrow$
$(1-tA)^{-1}=\displaystyle\sum_{k=0}^\infty (tA)^k$ konvergiert für $t<1/\pi(A)$
}
\end{block}}
\end{column}
\end{columns}
\end{frame}
\egroup