diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/5/Makefile.inc | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/5/chapter.tex | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/5/spektralgelfand.tex | 190 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 1 |
4 files changed, 194 insertions, 1 deletions
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 03ce407..00c8337 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -22,6 +22,8 @@ chapter5 = \ ../slides/5/reellenormalform.tex \ ../slides/5/cayleyhamilton.tex \ \ + ../slides/5/spektralgelfand.tex \ + \ ../slides/5/stoneweierstrass.tex \ ../slides/5/potenzreihenmethode.tex \ ../slides/5/logarithmusreihe.tex \ diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 31c6d25..6f3228d 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -19,7 +19,7 @@ \folie{5/jordan.tex} \folie{5/reellenormalform.tex} \folie{5/cayleyhamilton.tex} - +\folie{5/spektralgelfand.tex} \folie{5/stoneweierstrass.tex} \folie{5/potenzreihenmethode.tex} \folie{5/logarithmusreihe.tex} diff --git a/vorlesungen/slides/5/spektralgelfand.tex b/vorlesungen/slides/5/spektralgelfand.tex new file mode 100644 index 0000000..9342cd6 --- /dev/null +++ b/vorlesungen/slides/5/spektralgelfand.tex @@ -0,0 +1,190 @@ +% +% 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 diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index e398f84..a46cd04 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -112,6 +112,7 @@ % XXX \folie{5/gelfandradius.tex} % XXX Spektralradius % XXX \folie{5/spektralradius.tex} +\folie{5/spektralgelfand.tex} % XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) % XXX \folie{5/satzvongelfand.tex} % XXX Logarithmus |