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-rw-r--r-- | vorlesungen/slides/5/motivation.tex | 67 |
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diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex new file mode 100644 index 0000000..b0a1d82 --- /dev/null +++ b/vorlesungen/slides/5/motivation.tex @@ -0,0 +1,67 @@ +% +% movitation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Motivation} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Matrix $A$ analysieren} +Matrix $A$ mit Minimalpolynom $m_A(X)$ vom +Grad $s$ +\end{block} +\uncover<2->{% +\begin{block}{Faktorisieren} +Minimalpolynom faktorisieren +\[ +m_A(X) += +(X-\mu_1)(X-\mu_2)\dots(X-\mu_s) +\] +\end{block}} +\uncover<3->{% +\begin{block}{Vertauschen} +$\sigma\in S_s$ eine Permutation von $1,\dots,s$ +ist +\begin{align*} +m_A(X) +&= +(X-\mu_{\sigma(1)}) +%(X-\mu_{\sigma(2)}) +\dots +(X-\mu_{\sigma(s)}) +\\ +0 +&= +(A-\mu_{\sigma(1)}) +%(A-\mu_{\sigma(2)}) +\dots +(A-\mu_{\sigma(s)}) +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Bedingung für $\mu_k$} +Permutation wählen so dass $\mu_k$ an erster Stelle steht: +\[ +0=(A-\mu_k) \prod_{i\ne k}(A-\mu_i) v +\] +für alle $v\in\Bbbk^n$. +\end{block}} +\uncover<5->{% +\begin{block}{Eigenwerte} +Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$ +gibt mit +\[ +(A-\mu)v = 0 +\Rightarrow Av = {\color{red}\mu} v +\] +Eigenwertbedingung +\end{block}} +\end{column} +\end{columns} +\end{frame} |