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%
% intro.tex
%
% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\def\r{4}
\def\rad#1{
\begin{scope}[rotate=#1]
\fill[color=blue!20] (0,0) -- (-60:\r) arc (-60:60:\r) -- cycle;
\fill[color=darkgreen!20] (0,0) -- (60:\r) arc (60:180:\r) -- cycle;
\fill[color=orange!20] (0,0) -- (180:\r) arc (180:300:\r) -- cycle;
\node[color=darkgreen] at (120:3.7) [rotate={#1+30}] {Algebra};
\node[color=orange] at (240:3.7) [rotate={#1+150}] {Analysis};
\node[color=blue] at (0:3.7) [rotate={#1-90}] {Zerlegung};
\end{scope}
}
\begin{frame}
\frametitle{Intro --- Matrizen}
\vspace{-25pt}
\begin{center}
\begin{tikzpicture}[>=latex,thick]
\only<1-8>{
\rad{-30}
\only<2->{ \node at (90:3.0) {Rechenregeln $A^2+A+I=0$}; }
\only<3->{ \node at (90:2.5) {Polynome $\chi_A(A)=0$, $m_A(A)=0$}; }
\only<4->{ \node at (90:2.0) {Projektion: $P^2=P$}; }
\only<5->{ \node at (90:1.5) {nilpotent: $N^k=0$}; }
}
\only<9-14>{
\rad{90}
\only<10->{ \node at (90:2.7) {Eigenbasis: $A=\sum \lambda_k P_k$}; }
\only<11->{ \node at (90:2.2) {Invariante Räume:
$AV\subset V, AV^\perp\subset V^\perp$}; }
}
\only<15-22>{
\rad{210}
\only<16->{ \node at (90:3.3) {Symmetrien}; }
\only<17->{ \node at (90:2.8) {Skalarprodukt erhalten:
$\operatorname{SO}(n)$}; }
\only<18->{ \node at (90:2.3) {Konstant $\Rightarrow$ Ableitung $=0$}; }
\only<19->{ \node at (90:1.5) {$\displaystyle \exp(A)
= \sum_{k=0}^\infty \frac{A^k}{k!}$};
}
}
\fill[color=red!20] (0,0) circle[radius=1.0];
\node at (0,0.25) {Matrizen};
\node at (0,-0.25) {$M_{m\times n}(\Bbbk)$};
\uncover<6->{
\node[color=darkgreen] at (4.3,3.4) [right] {Algebra};
\node at (4.3,2.2) [right] {\begin{minipage}{5cm}
\begin{itemize}
\item<6-> Algebraische Strukturen
\item<7-> Polynome, Teilbarkeit
\item<8-> Minimalpolynom
\end{itemize}
\end{minipage}};
}
\uncover<12->{
\node[color=blue] at (4.3,0.8) [right] {Zerlegung};
\node at (4.3,-0.4) [right] {\begin{minipage}{5cm}
\begin{itemize}
\item<12-> Eigenvektoren, -räume
\item<13-> Projektionen, Drehungen
\item<14-> Invariante Unterräume
\end{itemize}
\end{minipage}};
}
\uncover<20->{
\node[color=orange] at (4.3,-1.8) [right] {Analysis};
\node at (4.3,-3.0) [right] {\begin{minipage}{6cm}
\begin{itemize}
\item<20-> Symmetrien
\item<21-> Matrix-DGL
\item<22-> Matrix-Potenzreihen
\end{itemize}
\end{minipage}};
}
\end{tikzpicture}
\end{center}
\end{frame}
\egroup
|
