<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%
% motivation.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\frametitle{Motivation}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.24\textwidth}
\begin{block}{Imaginäre Einheit}
\vspace{-15pt}
\begin{align*}
J &amp;= \begin{pmatrix} 0&amp;-1\\1&amp;0\end{pmatrix}
\\
p(X) &amp;= X^2 + 1
\\
p(J) &amp;= J^2 + I = 0
\end{align*}
\end{block}
\end{column}
\begin{column}{0.25\textwidth}
\uncover&lt;2-&gt;{%
\begin{block}{Wurzel $\sqrt{2}$}
\vspace{-15pt}
\begin{align*}
W&amp;=\begin{pmatrix}0&amp;2\\1&amp;0\end{pmatrix}
\\
p(X) &amp;= X^2-2
\\
p(W) &amp;= W^2-2I=0
\end{align*}
\end{block}}
\end{column}
\begin{column}{0.41\textwidth}
\uncover&lt;3-&gt;{%
\begin{block}{Drehmatrix}
\vspace{-15pt}
\begin{align*}
D&amp;=\begin{pmatrix}
\cos \frac{\pi}{1291} &amp; -\sin\frac{\pi}{1291}\\
\sin \frac{\pi}{1291} &amp;  \cos\frac{\pi}{1291}
\end{pmatrix}
\\
p(X)&amp;=
\ifthenelse{\boolean{presentation}}{\only&lt;-3&gt;{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
\only&lt;4-&gt;{X^2-2X\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
\\
p(D) &amp;= \ifthenelse{\boolean{presentation}}{\only&lt;-3&gt;{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
\only&lt;4-&gt;{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
\end{align*}
\end{block}}
\end{column}
\end{columns}
\vspace{-20pt}
\uncover&lt;5-&gt;{
\begin{block}{3D-Beispiel}
$p(x) = -x^3-5x^2+5x+1$
\[
\ifthenelse{\boolean{presentation}}{
\only&lt;5-8&gt;{
A=
\begin{pmatrix*}[r]
-5&amp;-1&amp;1\\
-5&amp;-2&amp;3\\
-1&amp;-1&amp;2
\end{pmatrix*}}
\only&lt;6-8&gt;{
\quad\Rightarrow\quad}}{}
\uncover&lt;6-&gt;{
-
\only&lt;-9&gt;{A^3}\only&lt;10-&gt;{
\begin{pmatrix*}[r]
-169&amp;-35&amp;35\\
-185&amp;-39&amp;40\\
 -45&amp;-10&amp;11
\end{pmatrix*}}
-5
\only&lt;-8&gt;{A^2}\only&lt;9-&gt;{
\begin{pmatrix*}[r]
29&amp;6&amp;-6\\
32&amp;6&amp;-5\\
 8&amp;1&amp; 0
\end{pmatrix*}}
+5
\only&lt;-7&gt;{A}\only&lt;8-&gt;{
\begin{pmatrix*}[r]
-5&amp;-1&amp;1\\
-5&amp;-2&amp;3\\
-1&amp;-1&amp;2
\end{pmatrix*}}
+
\only&lt;-6&gt;{I}\only&lt;7-&gt;{
\begin{pmatrix*}[r]
1&amp;0&amp;0\\
0&amp;1&amp;0\\
0&amp;0&amp;1
\end{pmatrix*}}
}
\uncover&lt;11-&gt;{=0}
\]
\end{block}}
\vspace{-10pt}
\uncover&lt;12-&gt;{%
{\usebeamercolor[fg]{title}$\Rightarrow$
Rechenregeln von Matrizen können durch Polynome ausgedrückt werden}
}
\end{frame}
</pre></body></html>