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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-02-28 22:37:26 +0100 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-02-28 22:37:26 +0100 |
commit | cddeab74130c3814acd49c8ac7d03041b2a7b85d (patch) | |
tree | b7fe6247083774badfd63e54467d2db340d23462 /vorlesungen/slides/3/einsetzen.tex | |
parent | typos (diff) | |
download | SeminarMatrizen-cddeab74130c3814acd49c8ac7d03041b2a7b85d.tar.gz SeminarMatrizen-cddeab74130c3814acd49c8ac7d03041b2a7b85d.zip |
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diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex new file mode 100644 index 0000000..936100d --- /dev/null +++ b/vorlesungen/slides/3/einsetzen.tex @@ -0,0 +1,50 @@ +% +% einsetzen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Matrix in ein Polynom einsetzen} +\vspace{-10pt} +\[ +\begin{array}{rcrcrcrcrcrcr} +p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\ +\bigg\downarrow\hspace*{4pt} & & +\bigg\downarrow\hspace*{4pt} & & +\bigg\downarrow\hspace*{10pt} & & & & +\bigg\downarrow\hspace*{4pt} & & +\bigg\downarrow\hspace*{2pt} & & +\bigg\downarrow\hspace*{0pt} \\ +p(A)&=&a_nA^n&+&a_{n-1}A^{n-1}&+&\dots&+&a_2A^2&+&a_1A&+&a_0 I +\end{array} +\] +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Nilpotente Matrizen} +$p(X) = (X-a)^n$ +\[ +p(A) = 0 +\quad\Rightarrow\quad +\text{$A$ ist nilpotent} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Eigenwerte} +$p(X) = (X-\lambda_1)(X-\lambda_2)$,\\ +$A$ eine $2\times 2$-Matrix +\[ +p(A)=0\quad\Rightarrow\quad +\left\{ +\begin{aligned} +&\text{$A-\lambda_1I$ ist singulär}\\ +&\text{$A-\lambda_2I$ ist singulär} +\end{aligned} +\right. +\] +\end{block} +\end{column} +\end{columns} + +\end{frame} |