%
% vergleich.tex -- slide template
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Vergleich}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick]

\def\a{1.2}    \def\b{0.35}
\def\c{0.5}    \def\d{1.25}
\def\r{4}

\coordinate (u) at (3.5,0);
\coordinate (v) at (2.5,0);

\coordinate (Au) at ({3.5*\a},{3.5*\c});
\coordinate (Av) at ({2.5*\a},{2.5*\c});

\uncover<2->{
	\begin{scope}
	\clip (0,0) rectangle (5,5);
	\fill[color=red!20] (0,0) circle[radius=4];
	\end{scope}
	\node[color=red] at (0,4) [below right] {$\mathbb{R}^n$};

	\fill[color=blue!40,opacity=0.5] (0,0) -- ({\a*\r},{\c*\r})
		-- plot[domain=0:90,samples=100]
		({\r*(\a*cos(\x)+\b*sin(\x))},{\r*(\c*cos(\x)+\d*sin(\x))})
		-- ({\b*\r},{\d*\r}) -- cycle;
	\node[color=blue] at ({\r*\b},{\r*\d}) [below right] {$A\mathbb{R}^n$};
}

\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}];
\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}];

\uncover<3->{
	\fill[color=darkgreen!30,opacity=0.5]
		(0,0) rectangle ({3.5*\a},{3.5*\c});
	\draw[color=white,line width=0.7pt]
		({3.5*\a},0) -- ({3.5*\a},{3.5*\c}) -- (0,{3.5*\c});
}

\uncover<2->{
	\draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\a},{\r*\c});
	\draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\b},{\r*\d});

	\draw[->,color=red,line width=1.4pt] (0,0) -- (4,0);
	\draw[->,color=red,line width=1.4pt] (0,0) -- (0,4);
}

\draw[color=darkgreen,line width=2pt] (u) -- (v);
\fill[color=darkgreen] (u) circle[radius=0.08];
\fill[color=darkgreen] (v) circle[radius=0.08];

\node[color=darkgreen] at (u) [below right] {$u$};
\node[color=darkgreen] at (v) [below left] {$v$};
\node[color=darkgreen] at ($0.5*(u)+0.5*(v)$) [above] {$v\le u$};

\uncover<3->{
	\draw[color=darkgreen,line width=2pt] (Au) -- (Av);
	\fill[color=darkgreen] (Au) circle[radius=0.08];
	\fill[color=darkgreen] (Av) circle[radius=0.08];

	\node[color=darkgreen] at (Au) [above left] {$Au$};
	\node[color=darkgreen] at (Av) [above left] {$Av$};

	\node[color=darkgreen] at ($0.5*(Au)+0.5*(Av)$)
		[below,rotate={atan(\c/\a)}] {$Av<Au$};
}

\end{tikzpicture}
\end{center}
\end{column}
\begin{column}{0.48\textwidth}
\begin{block}{Satz}
$u\ge v\ge 0$ \uncover<2->{und $A > 0$}\uncover<3->{ $\Rightarrow$ $Au>Av$}
\end{block}
\uncover<4->{%
\begin{block}{intuitiv}
$A>0$ befördert $\ge$ zu $>$
\end{block}}
\uncover<5->{%
\begin{proof}[Beweis]
$d=u-v\ge 0$
\begin{align*}
(Ad)_i
\uncover<6->{=
\sum_{j}
\underbrace{a_{ij}}_{>0}d_j}
\uncover<7->{>
0}
\uncover<8->{\quad\Rightarrow\quad
Au > Av}
\end{align*}
\uncover<7->{da mindestens ein $d_j>0$ ist}
\end{proof}}
\uncover<9->{%
\begin{block}{Korollar}
$A>0$ und $d\ge 0$ $\Rightarrow$ $Ad > 0$
\end{block}}
\end{column}
\end{columns}
\end{frame}
\egroup