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diff --git a/vorlesungen/slides/5/plan.tex b/vorlesungen/slides/5/plan.tex new file mode 100644 index 0000000..23b1b93 --- /dev/null +++ b/vorlesungen/slides/5/plan.tex @@ -0,0 +1,198 @@ +% +% plan.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.5,0} +\definecolor{darkred}{rgb}{0.8,0.0,0} +\begin{frame}[t] +\frametitle{Was ist $f(A)$?} +\vspace{-5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<7->{ + \fill[color=blue!20] (-1.5,0.7) rectangle (11.5,3.8); +} + +\uncover<4->{ + \fill[color=darkgreen!20] (-1.5,-0.7) rectangle (11.5,0.7); +} + +\uncover<12->{ + \fill[color=darkred!20] (-1.5,-0.7) rectangle (11.5,-3.8); +} + +\begin{scope}[xshift=-1cm] +\node at (0,0) [left] {$A$}; +\end{scope} + +%\foreach \x in {1,...,20}{ +% \only<\x>{ \node at (-1,3) {\x}}; +%} + +% +% Blauer Ast +% + +\uncover<2->{ + \draw[->,color=blue,shorten <= 0.3cm, shorten >= 0.0cm] + (-1.2,0) -- (0,1.3); + + \begin{scope}[xshift=0cm,yshift=1.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6); + \draw[color=blue] (0,-0.6) rectangle (3.4,0.6); + \node at (0,0) [right] {$\begin{aligned} + f&=p\in\mathbb{R}[X]\\ + f(A)&=p(A) + \end{aligned} + $}; + \end{scope} +} + +\uncover<7->{ + \draw[->,color=blue] (1.8,2.1) -- (3.6,3); + + \begin{scope}[xshift=3.6cm,yshift=3cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6); + \draw[color=blue] (0,-0.6) rectangle (3.7,0.6); + \node at (0,0) [right] {\begin{minipage}{3cm}\raggedright + $f$ durch $p_n\in\mathbb{R}[X]$\\ + approximieren + \end{minipage}}; + \end{scope} +} + +\uncover<8->{ + \draw[->,color=blue] (7.3,3) -- (9.5,1.9); + + \begin{scope}[xshift=7.6cm,yshift=1.5cm] + \fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4); + \draw[color=blue] (0,-0.35) rectangle (3.8,0.4); + \node at (0,0) [right] {$\displaystyle f(A) = \lim_{n\to\infty}p_n(A)$}; + \end{scope} +} + +\uncover<9->{ + \node[color=blue] at (3.6,1.6) [right] {\begin{minipage}{4cm} + \raggedright + Konvergenz $p_n\to f$\\ + auf Spektrum $\operatorname{Sp}(A)\subset\mathbb{R}$ + \end{minipage}}; +} + +\uncover<11->{ + \node[color=blue] at (-1.5,3.8) [below right] + {$A$ symmetrisch: $A=A^*$}; +} +\uncover<10->{ + \node[color=blue] at (11.5,3.8) [below left] {$A$ diagonalisierbar}; +} + +% +% Roter Ast +% + +\uncover<12->{ + \draw[->,color=darkred,shorten <= 0.3cm, shorten >= 0.0cm] (-1.2,0) -- (0,-1.3); + + \begin{scope}[xshift=0cm,yshift=-1.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6); + \draw[color=darkred] (0,-0.6) rectangle (3.4,0.6); + \node at (0,0) [right] {$\begin{aligned} + f&=p\in\mathbb{C}[Z,\overline{Z}]\\ + f(A)&=p(A,A^*) + \end{aligned}$}; + \end{scope} +} + +\uncover<13->{ + \node[color=darkred] at (1.7,-2.1) [below left] + {Für $|Z|^2 = Z\overline{Z}$}; +} + +\uncover<14->{ + \draw[->,color=darkred] (1.8,-2.1) -- (3.6,-3); + + \begin{scope}[xshift=3.6cm,yshift=-3cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6); + \draw[color=darkred] (0,-0.6) rectangle (3.7,0.6); + \node at (0,0) [right] {\begin{minipage}{3.5cm}\raggedright + $f$ durch $q_n\in\mathbb{C}[Z,\overline{Z}]$\\ + approximieren + \end{minipage}}; + \end{scope} +} + +\uncover<15->{ + \draw[->,color=darkred] (7.3,-3) -- (9.5,-1.85); + + \begin{scope}[xshift=7.6cm,yshift=-1.5cm] + \fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4); + \draw[color=darkred] (0,-0.35) rectangle (3.8,0.4); + \node at (0,0) [right] + {$\displaystyle f(A) = \lim_{n\to\infty}q_n(A,A^*)$}; + \end{scope} +} + +\uncover<16->{ + \node[color=darkred] at (3.6,-1.8) [right] {\begin{minipage}{4cm} + \raggedright + Konvergenz $p_n\to f$\\ + auf $\operatorname{Sp}(A)\cup\operatorname{Sp}(A^*)$ + \end{minipage}}; +} + +\uncover<17->{ + \node[color=darkred] at (11.5,-3.8) [above left] {% + \begin{minipage}{3.5cm}\raggedleft + nur sinnvoll definiert wenn + $AA^*=A^*A$ + \end{minipage}}; +} + +\uncover<18->{ + \node[color=darkred] at (-1.5,-3.8) [above right] + {$A$ normal: $AA^*=A^*A$}; +} + +% +% Grüner Ast +% + +\uncover<3->{ + \draw[->,color=darkgreen,shorten <= 0.0cm, shorten >= 0.0cm] + (-1,0) -- (0,0); + + \begin{scope}[xshift=0cm,yshift=0cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6); + \draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6); + \node at (0,0) [right] {$\displaystyle + f(z)=\sum_{k=0}^\infty a_kz^k$}; + \end{scope} +} + +\uncover<5->{ + \node[color=darkgreen] at (5.9,0) [above] {$f(z)$ analytisch!}; +} +\uncover<6->{ + \node[color=darkgreen] at (5.9,0) [below] + {$\varrho(A)<\text{Konvergenzradius}$}; +} + +\uncover<4->{ + \draw[->,color=darkgreen] (2.9,0) -- (8.5,0); + + \begin{scope}[xshift=8.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6); + \draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6); + \node at (0,0) [right] {$\displaystyle + f(A)=\sum_{k=0}^\infty a_kA^k$}; + \end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup |