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| -rw-r--r-- | vorlesungen/slides/3/inverse.tex | 89 |
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diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex new file mode 100644 index 0000000..28716bc --- /dev/null +++ b/vorlesungen/slides/3/inverse.tex @@ -0,0 +1,89 @@ +% +% inverse.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Inverse Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.24\textwidth} +\begin{block}{Imaginäre Einheit} +\vspace{-15pt} +\begin{align*} +J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix} +\\ +0&= +J^2 + I +\\ +0&= +J+J^{-1} +\\ +J^{-1}&=-J +\end{align*} +\end{block} +\end{column} +\begin{column}{0.25\textwidth} +\uncover<2->{% +\begin{block}{Wurzel $\sqrt{2}$} +\vspace{-15pt} +\begin{align*} +W&=\begin{pmatrix}0&2\\1&0\end{pmatrix} +\\ +0 &= X^2-2 +\\ +0 &= W-2W^{-1} +\\ +W^{-1}&=\frac12 W +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.41\textwidth} +\uncover<3->{% +\begin{block}{Drehmatrix} +\vspace{-15pt} +\begin{align*} +D&=\begin{pmatrix} +\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\ +\sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} +\end{pmatrix} +\\ +0 &= \only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}} +\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +0 &= \only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}} +\only<4->{D-2\cos\frac{\pi\mathstrut}{1291\mathstrut}+D^{-1}} +\\ +D^{-1} +&= \only<-3>{-D^{1290}\phantom{+\frac{\mathstrut}{\mathstrut}}}% +\only<4->{-D+2I\cos\frac{\pi\mathstrut}{1291\mathstrut}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-25pt} +\uncover<5->{ +\begin{block}{3D-Beispiel} +$p(x) = -x^3-5x^2+5x+1$ +\[ +A= +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*} +\quad\Rightarrow\quad +A^{-1} += +A^2+5A-5I += +\begin{pmatrix*}[r] +-1& 1&-1\\ + 7&-9&10\\ + 3&-4& 5 +\end{pmatrix*} +\] +\end{block}} +\vspace{-10pt} + +\end{frame} |