diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/5/Makefile.inc | 7 | ||||
| -rw-r--r-- | vorlesungen/slides/5/chapter.tex | 7 | ||||
| -rw-r--r-- | vorlesungen/slides/5/charpoly.tex | 68 | ||||
| -rw-r--r-- | vorlesungen/slides/5/dimension.tex | 68 | ||||
| -rw-r--r-- | vorlesungen/slides/5/folgerungen.tex | 31 | ||||
| -rw-r--r-- | vorlesungen/slides/5/kernbild.tex | 68 | ||||
| -rw-r--r-- | vorlesungen/slides/5/ketten.tex | 78 | ||||
| -rw-r--r-- | vorlesungen/slides/5/motivation.tex | 63 | ||||
| -rw-r--r-- | vorlesungen/slides/5/nilpotent.tex | 176 |
9 files changed, 566 insertions, 0 deletions
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index d690514..d081b29 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -5,5 +5,12 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapter5 = \ + ../slides/5/motivation.tex \ + ../slides/5/charpoly.tex \ + ../slides/5/kernbild.tex \ + ../slides/5/ketten.tex \ + ../slides/5/dimension.tex \ + ../slides/5/folgerungen.tex \ + ../slides/5/nilpotent.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 884732f..4bcee8e 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,3 +3,10 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % +folie{5/motivation.tex} +folie{5/charpoly.tex} +folie{5/kernbild.tex} +folie{5/ketten.tex} +folie{5/dimension.tex} +folie{5/folgerungen.tex} +folie{5/nilpotent.tex} diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex new file mode 100644 index 0000000..1211b43 --- /dev/null +++ b/vorlesungen/slides/5/charpoly.tex @@ -0,0 +1,68 @@ +% +% charpoly.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Charakteristisches Polynom über $\mathbb{C}$} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Eigenwerte} +Nur diejenigen $\mu$ kommen in Frage, für die +$A-\mu I$ singulär ist: +\[ +\chi_{A}(\mu) += +\det (A-\mu I) = 0 +\] +$\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$ +\end{block} +\begin{block}{Zerlegung in Linearfaktoren} +$\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$: +\[ +\chi_A(X) += +(X-\mu_1)\dots (X-\mu_n) +\] +\end{block} +\begin{block}{Fundamentalsatz der Algebra} +Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in +Linearfaktoren +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Minimalpolynom} +Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen +\end{block} +\begin{proof}[Beweis] +\begin{enumerate} +\item +$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$ +\item +$A-\lambda I$ ist regulär +\end{enumerate} +\begin{align*} +&\Rightarrow& +m_A(A)&=0 +\\ +&& +(A-\lambda)^{-1}m_A(A) &=0 +\\ +&& +\prod_{i\in I}(A-\mu_i)&=0, +\end{align*} +d.~h.~\( +\displaystyle +\overline{m}_A(X) += +\prod_i{i\in I}(X-\mu_i) +\in +\mathbb{C}[X] +\) +\end{proof} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/dimension.tex b/vorlesungen/slides/5/dimension.tex new file mode 100644 index 0000000..ff687b3 --- /dev/null +++ b/vorlesungen/slides/5/dimension.tex @@ -0,0 +1,68 @@ +% +% dimension.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Dimension von $\mathcal{K}^k(f)$ und $\mathcal{J}^k(f)$} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\pfad{ + (0,0) -- (1,0.3) -- (2,0.9) + -- + (4,2.4) -- (5,2.7) -- (6,3.3) + -- + (8,3.7) -- (9,4) -- (10,4) -- (11,4) -- (12,4) +} + +\fill[color=darkgreen!20] \pfad -- (12,0) -- cycle; +\fill[color=orange!20] \pfad -- (12,6) -- (0,6) -- cycle; + +\fill[color=darkgreen!40] (9,0) -- (12,0) -- (12,4) -- (9,4) -- cycle; +\fill[color=orange!40] (9,4) -- (12,4) -- (12,6) -- (9,6) -- cycle; + +\node[color=orange] at (10.5,5) {$\mathcal{J}(f)$}; +\node[color=darkgreen] at (10.5,2) {$\mathcal{K}(f)$}; + +\node[color=orange] at (5.5,4.5) {$\mathcal{J}^k(f)\supset\mathcal{J}^{k+1}(f)$}; +\node[color=darkgreen] at (5.5,1.5) {$\mathcal{K}^k(f)\subset\mathcal{K}^{k+1}(f)$}; + +\draw[line width=1.4pt] \pfad; + +\draw[->] (-0.1,6) -- (12.5,6) coordinate[label={$k$}]; +\draw[->] (-0.1,0) -- (12.5,0) coordinate[label={$k$}]; +\node at (-0.1,6) [left] {$n$}; +\node at (-0.1,0) [left] {$0$}; +\foreach \x in {0,1,2,4,5,6,8,9,10,11,12}{ + \fill (\x,0) circle[radius=0.05]; + \fill (\x,6) circle[radius=0.05]; +} +\node at (0,0) [below] {$0$}; +\node at (1,0) [below] {$1$}; +\node at (2,0) [below] {$2$}; + +\node at (4,0) [below] {$k-1$}; +\node at (5,0) [below] {$k$}; +\node at (6,0) [below] {$k+1$}; + +\node at (8,0) [below] {$l-1$}; +\node at (9,0) [below] {$l$}; +\node at (10,0) [below] {$l+1$}; +\node at (11,0) [below] {$l+2$}; +\node at (12,0) [below] {$l+3$}; + +\fill (9,4) circle[radius=0.05]; + +\node[color=orange] at (-0.2,3) [rotate=90] {$\dim\mathcal{J}^k(f)$}; +\node[color=darkgreen] at (12.2,2) [rotate=-90] {$\dim\mathcal{K}^k(f)$}; + +\node[color=orange] at (9,5) [rotate=-90] {$\dim\mathcal{J}(f)$}; +\node[color=darkgreen] at (9,2) [rotate=-90] {$\dim\mathcal{K}(f)$}; + +\end{tikzpicture} +\end{center} + +\end{frame} diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex new file mode 100644 index 0000000..96efd7f --- /dev/null +++ b/vorlesungen/slides/5/folgerungen.tex @@ -0,0 +1,31 @@ +% +% folgerungen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Folgerungen} +\begin{columns}[t] +\begin{column}{0.48\textwidth} +\begin{block}{Zunahme} +Für alle $k<l$ gilt +\begin{align*} +\mathcal{J}^k(f) &\supsetneq \mathcal{J}^{k+1}(f) +\\ +\mathcal{K}^k(f) &\subsetneq \mathcal{K}^{k+1}(f) +\end{align*} +Für $k\ge l$ gilt +\begin{align*} +\mathcal{J}^k(f) &= \mathcal{J}^{k+1}(f) +\\ +\mathcal{K}^k(f) &= \mathcal{K}^{k+1}(f) +\end{align*} +Ausserdem ist $l\le n$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex new file mode 100644 index 0000000..f0bd6fa --- /dev/null +++ b/vorlesungen/slides/5/kernbild.tex @@ -0,0 +1,68 @@ +% +% kernbild.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kern und Bild} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Kern} +Lineare Abbildung $f\colon V\to V$ +\[ +\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\} +\] +\end{block} +\begin{block}{Kern von $A^k$} +\[ +\mathcal{K}^k(f) = \operatorname{ker} f^k +\] +\begin{align*} +\mathcal{K}^k(f) +&= +\{v\in V\;|\; f^{k}(v)=0\} +\\ +&\subset +\{v\in V\;|\; f^{k+1}(v)=0\} +\\ +&=\mathcal{K}^{k+1}(f) +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Bild} +Lineare Abbildung $f\colon V\to V$ +\[ +\operatorname{im}f += +\mathcal{J}(f) += +\{f(v)\;|\; v\in V\} +\] +\end{block} +\begin{block}{Bild von $A^k$} +\[ +\mathcal{J}^k(f) = \operatorname{im}f^k +\] +\begin{align*} +\mathcal{J}^k(f) +&= +\operatorname{im}f^k += +\operatorname{im}(f^{k}\circ f) +\\ +&= +\{f^{k-1} w\;|\; w = f(v)\} +\\ +&\subset +\{f^{k-1} w\;|\; w \in V\} +\\ +&\mathcal{J}^{k-1}(f) +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex new file mode 100644 index 0000000..759d964 --- /dev/null +++ b/vorlesungen/slides/5/ketten.tex @@ -0,0 +1,78 @@ +% +% ketten.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ketten von Unterräumen} +\begin{block}{Schachtelung} +Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt: +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcccc} +0 &=&\mathcal{K}^0(f) + &\subset&\mathcal{K}^1(f) + &\subset&\dots + &\subset&\mathcal{K}^k(f) + &\subset&\mathcal{K}^{k+1}(f) + &\subset&\dots + &\subset&\displaystyle\bigcup_{k=0}^\infty \mathcal{K}^k(f) + &=:&\mathcal{K}(f) +\\[14pt] +\Bbbk^n &=&\mathcal{J}^0(f) + &\supset&\mathcal{J}^1(f) + &\supset&\dots + &\supset&\mathcal{J}^{k}(f) + &\supset&\mathcal{J}^{k+1}(f) + &\supset&\dots + &\supset&\displaystyle\bigcap_{k=0}^\infty \mathcal{J}^k(f) + &=:&\mathcal{J}(f) +\end{array} +\] +\end{block} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Abildung der Kerne} +\vspace{-10pt} +\begin{align*} +f \mathcal{K}^k(f) +&= +\{f(v)\;|\; f^k(v) = 0\} +\\ +&\subset +\{ v\;|\; f^{k+1}(v)=0\} +\\ +&= +\mathcal{K}^{k+1}(f) +\\ +\Rightarrow +f\mathcal{K}(f)&= f\mathcal{K}(f) +\quad\text{invariant} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Abbildung der Bild} +\vspace{-10pt} +\begin{align*} +f\mathcal{J}^k(f) +&= +\{f(f^{k}(v))\;|\; v\in V\} +\\ +&= +\{f^{k+1}(v)\;|\; v\in V\} +\\ +&= +\mathcal{J}^{k+1}(f) +\\ +\Rightarrow +f\mathcal{J}(f)&= \mathcal{J}(f) +\quad\text{invariant} +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex new file mode 100644 index 0000000..4e8142d --- /dev/null +++ b/vorlesungen/slides/5/motivation.tex @@ -0,0 +1,63 @@ +% +% 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} +\begin{block}{Faktorisieren} +Minimalpolynom faktorisieren +\[ +m_A(X) += +(X-\mu_1)(X-\mu_2)\dots(X-\mu_s) +\] +\end{block} +\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} +\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} +\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} diff --git a/vorlesungen/slides/5/nilpotent.tex b/vorlesungen/slides/5/nilpotent.tex new file mode 100644 index 0000000..9b7ded1 --- /dev/null +++ b/vorlesungen/slides/5/nilpotent.tex @@ -0,0 +1,176 @@ +% +% nilpotent.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\feld#1{ + \fill[color=red!20] (#1,0) rectangle ({#1+1},12); +} +\begin{frame}[t] +\frametitle{$\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ für nilpotente Matrizen} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.42\textwidth} +Matrix mit dem dargestellten Verlauf von +${\color{red}\dim\mathcal{K}^k(A)}$ +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.42] + +\only<2->{ + \feld{0} + \feld{1} + \feld{2} + \feld{3} +} +\only<2->{ \feld{4} } +\only<3->{ \feld{5} } +\only<2->{ \feld{6} } +\only<3->{ \feld{7} } +\only<4->{ \feld{8} } +\only<5->{ \feld{9} } +\only<6->{ \feld{10} } +\only<7->{ \feld{11} } + +\only<1>{ \node at (6,0) [below] {$k=0$}; } +\only<2>{ \node at (6,0) [below] {$k=1$}; } +\only<3>{ \node at (6,0) [below] {$k=2$}; } +\only<4>{ \node at (6,0) [below] {$k=3$}; } +\only<5>{ \node at (6,0) [below] {$k=4$}; } +\only<6>{ \node at (6,0) [below] {$k=5$}; } +\only<7>{ \node at (6,0) [below] {$k=6$}; } + +\draw (0,0) rectangle (12,12); +\only<1>{ + \foreach \x in {1,...,12}{ + \node at ({\x-0.5},{12-\x+0.5}) {$1$}; + } +} +\only<2->{ + \foreach \x in {1,...,12}{ + \node at ({\x-0.5},{12-\x+0.5}) {$0$}; + } +} +\only<2>{ + \foreach \x in {7,...,11}{ + \node at ({\x+0.5},{12-\x+0.5}) {$1$}; + } +} +\only<3->{ + \foreach \x in {7,...,11}{ + \node at ({\x+0.5},{12-\x+0.5}) {$0$}; + } +} +\only<3>{ + \foreach \x in {8,...,11}{ + \node at ({\x+0.5},{13-\x+0.5}) {$1$}; + } +} +\only<4->{ + \foreach \x in {8,...,11}{ + \node at ({\x+0.5},{13-\x+0.5}) {$0$}; + } +} +\only<4>{ + \foreach \x in {9,...,11}{ + \node at ({\x+0.5},{14-\x+0.5}) {$1$}; + } +} +\only<5->{ + \foreach \x in {9,...,11}{ + \node at ({\x+0.5},{14-\x+0.5}) {$0$}; + } +} +\only<5>{ + \foreach \x in {10,...,11}{ + \node at ({\x+0.5},{15-\x+0.5}) {$1$}; + } +} +\only<6->{ + \foreach \x in {10,...,11}{ + \node at ({\x+0.5},{15-\x+0.5}) {$0$}; + } +} +\only<6>{ + \foreach \x in {11,...,11}{ + \node at ({\x+0.5},{16-\x+0.5}) {$1$}; + } +} +\only<7->{ + \foreach \x in {11,...,11}{ + \node at ({\x+0.5},{16-\x+0.5}) {$0$}; + } +} +\draw[line width=0.1pt] + (0,11) -- (2,11) -- (2,9) -- (4,9) -- (4,6) -- (12,6); +\draw[line width=0.1pt] + (1,12) -- (1,10) -- (3,10) -- (3,8) -- (6,8) -- (6,0); +\only<2>{ + \node at (5.5,7.5) {$1$}; +} +\only<3->{ + \node at (5.5,7.5) {$0$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.56\textwidth} +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\pfad{ + (0,0) -- (1,3) -- (2,4) -- (3,4.5) -- (4,5) -- (5,5.5) -- (6,6) +} +\fill[color=orange!20] \pfad -- (0,6) -- cycle; +\fill[color=darkgreen!20] \pfad -- (6,0) -- cycle; +\foreach \y in {0.5,1,...,5.75}{ + \draw[line width=0.1pt] (0,\y) -- (6,\y); +} +\draw[line width=1.4pt] \pfad; +\draw[->] (-0.1,6) -- (6.5,6); \node at (-0.1,6) [left] {$n$}; +\draw[->] (-0.1,0) -- (6.5,0); \node at (-0.1,0) [left] {$0$}; +\fill (0,0) circle[radius=0.05]; +\fill (1,3) circle[radius=0.05]; +\fill (2,4) circle[radius=0.05]; +\fill (3,4.5) circle[radius=0.05]; +\fill (4,5) circle[radius=0.05]; +\fill (5,5.5) circle[radius=0.05]; +\fill (6,6) circle[radius=0.05]; +\only<1>{ + \fill[color=red] (0,0) circle[radius=0.08]; +} +\only<2>{ + \fill[color=red] (1,3) circle[radius=0.08]; + \draw[color=red] (0,3) -- (1,3); + \node[color=red] at (0,3) [left] {$6$}; +} +\only<3>{ + \fill[color=red] (2,4) circle[radius=0.08]; + \draw[color=red] (0,4) -- (2,4); + \node[color=red] at (0,4) [left] {$8$}; +} +\only<4>{ + \fill[color=red] (3,4.5) circle[radius=0.08]; + \draw[color=red] (0,4.5) -- (3,4.5); + \node[color=red] at (0,4.5) [left] {$9$}; +} +\only<5>{ + \fill[color=red] (4,5.0) circle[radius=0.08]; + \draw[color=red] (0,5.0) -- (4,5.0); + \node[color=red] at (0,5.0) [left] {$10$}; +} +\only<6>{ + \fill[color=red] (5,5.5) circle[radius=0.08]; + \draw[color=red] (0,5.5) -- (5,5.5); + \node[color=red] at (0,5.5) [left] {$11$}; +} +\only<7>{ + \fill[color=red] (6,6.0) circle[radius=0.08]; +} +\draw[color=white] (-0.7,0) -- (-0.7,6); +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup |