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Diffstat (limited to 'vorlesungen/slides/5/normalnilp.tex')
| -rw-r--r-- | vorlesungen/slides/5/normalnilp.tex | 229 |
1 files changed, 229 insertions, 0 deletions
diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex index a7af682..9457136 100644 --- a/vorlesungen/slides/5/normalnilp.tex +++ b/vorlesungen/slides/5/normalnilp.tex @@ -3,6 +3,235 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\sx{1.9} +\def\sy{0.6} +\def\punkt#1#2#3{ + \foreach \y in {0,...,#2}{ + } +} +\def\block#1#2{ + \fill[rounded corners=2pt,color=white] + ({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05}); + \draw[rounded corners=2pt] + ({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05}); +} +\def\teilmenge#1#2#3{ + \fill[rounded corners=2pt,color=white] + ({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); + \draw[rounded corners=2pt,color=gray] + ({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); +} +\def\rot#1#2#3{ + \fill[rounded corners=2pt,color=red!20] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); + \draw[rounded corners=2pt,color=red] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); +} +\def\hellblau#1#2#3{ + \fill[rounded corners=2pt,color=blue!20] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); + \draw[rounded corners=2pt,color=blue!40] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); +} +\def\punkt#1#2{ + \fill[color=blue] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08]; +} +\def\bildpunkt#1#2{ + \fill[color=blue!40] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08]; +} +\def\pfeil#1#2#3{ + \draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.1cm] + ({-#1*\sx},{(#2-0.5)*\sy}) + -- + ({-(#1-1)*\sx},{(#3-0.5)*\sy}) ; +} \begin{frame}[t] \frametitle{Normalform einer nilpotenten Matrix} +{\usebeamercolor[fg]{title}$A^l=0$ $\Rightarrow$ finde eine ``gute'' Basis} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\vspace{-25pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=darkgreen!20,rounded corners=2pt] + ({-3*\sx+0.35},0) -- (-0.35,0) + -- + ({-1*\sx+0.35},{4*\sy}) -- ({-1*\sx-0.35},{4*\sy}) + -- + ({-2*\sx+0.35},{7*\sy}) -- ({-2*\sx-0.35},{7*\sy}) + -- + ({-3*\sx+0.35},{8*\sy}) -- cycle; + +\block{0}{0} + +\block{1}{4} +\uncover<10->{ + \rot{1}{0}{1} + \node[color=red] at ({-1*\sx-0.28},{0.5*\sy}) [left] {$\mathcal{C}_{l-2}$}; +} +\uncover<8->{ + \hellblau{1}{1}{3} +} +\uncover<4->{ + \hellblau{1}{3}{4} +} + +\block{2}{7} +\uncover<4->{ + \hellblau{2}{6}{7} +} +\uncover<6->{ + \rot{2}{4}{6} + \node[color=red] at ({-2*\sx-0.28},{5*\sy}) [left] {$\mathcal{C}_{l-1}$}; +} +\teilmenge{2}{0}{4} + +\block{3}{8} +\uncover<2->{ + \rot{3}{7}{8} + \node[color=red] at ({-3*\sx-0.28},{7.5*\sy}) [left] {$\mathcal{C}_l$}; +} +\teilmenge{3}{0}{7} + +\uncover<3->{ + \punkt{3}{8} +} +\uncover<4->{ + \pfeil{3}{8}{7} + \bildpunkt{2}{7} + \pfeil{2}{7}{4} + \bildpunkt{1}{4} +} + +\uncover<7->{ + \punkt{2}{5} + \punkt{2}{6} +} +\uncover<8->{ + \pfeil{2}{5}{2} + \bildpunkt{1}{3} + \pfeil{2}{6}{3} + \bildpunkt{1}{2} +} + +\uncover<11->{ +\punkt{1}{1} +} + +\node at ({-3*\sx},0) [below] {$\mathcal{K}^l(A)\mathstrut$}; +\node at ({-2*\sx},0) [below] {$\mathcal{K}^{l-1}(A)\mathstrut$}; +\node at ({-1.45*\sx},0) [below] {$\dots\mathstrut$}; +\node at ({-1*\sx},0) [below] {$\mathcal{K}^1(A)\mathstrut$}; +\node at ({-0*\sx},0) [below] {$0=\mathcal{K}^0(A)\mathstrut$}; +\node[color=gray] at ({-2*\sx},{2*\sy}) [rotate=90] {$\mathcal{K}^1(A)$}; +\node[color=gray] at ({-3*\sx},{3.5*\sy}) [rotate=90] {$\mathcal{K}^{l-1}(A)$}; +\foreach \x in {0,1,2}{ + \draw[->,shorten >= 0.1cm, shorten <= 0.1cm] + ({-(\x+1)*\sx},{8.7*\sy}) -- ({-(\x)*\sx},{8.7*\sy}); + \node at ({-(\x+0.5)*\sx},{8.7*\sy}) [above] {$A$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\vspace{-30pt} +\begin{enumerate} +\item<2-> \( + \mathcal{K}^l(A)=\mathcal{K}^{l-1}\oplus {\color{red}\mathcal{C}_l} + \) +\item<3-> \( + {\color{blue}b_1}\in{\color{red}\mathcal{C}_l} + \) +\item<4-> \( + \mathcal{B}_l + = + \{{\color{blue}b_1},{\color{blue!40}Ab_1},{\color{blue!40}A^2b_1},\dots, + {\color{blue!40}A^{l-1}b_1}\} + \) +\item<5-> \( + \mathcal{K}^{l-1}(A) + = + \mathcal{K}^{l-2}(A) + \oplus + {\color{red}\mathcal{C}_{l-1}} + \oplus + {\color{blue}A\mathcal{C}_l} + \) +\item<6-> \( + {\color{blue}b_2},{\color{blue}b_3}\in{\color{red}\mathcal{C}_{l-1}} + \) +\item<7-> \( + \mathcal{B}_{l-1} + = + \{ + {\color{blue}b_2},{\color{blue}b_3}, + {\color{blue!40}Ab_2}, {\color{blue!40}Ab_3},\dots + \} + \) +\item<8-> \dots +\end{enumerate} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] + +\uncover<2-4>{ + \fill[color=red!20] (2,0) rectangle (3,8); +} +\uncover<4->{ + \fill[color=blue!20] (0,6) rectangle (2,8); +} +\uncover<5->{ + \fill[color=red!20] (2,5) rectangle (3,8); + \node[color=blue] at (2.5,6.5) {$1$}; + \node[color=blue] at (1.5,7.5) {$1$}; + \node[color=gray] at (0.5,7.5) {$0$}; + \node[color=gray] at (1.5,6.5) {$0$}; + \node[color=gray] at (2.5,5.5) {$0$}; + \draw[color=gray] (0.05,5.05) rectangle (2.95,7.95); +} + +\uncover<6-8>{ + \fill[color=red!20] (4,0) rectangle (5,8); + \fill[color=red!20] (6,0) rectangle (7,8); +} +\uncover<8->{ + \fill[color=blue!20] (3,4) rectangle (4,5); + \fill[color=blue!20] (5,2) rectangle (6,3); +} +\uncover<9->{ + \fill[color=red!20] (4,3) rectangle (5,5); + \node[color=blue] at (4.5,4.5) {$1$}; + \node[color=gray] at (3.5,4.5) {$0$}; + \node[color=gray] at (4.5,3.5) {$0$}; + \draw[color=gray] (3.05,3.05) rectangle (4.95,4.95); + \fill[color=red!20] (6,1) rectangle (7,3); + \node[color=blue] at (6.5,2.5) {$1$}; + \node[color=gray] at (5.5,2.5) {$0$}; + \node[color=gray] at (6.5,1.5) {$0$}; + \draw[color=gray] (5.05,1.05) rectangle (6.95,2.95); +} + +\uncover<10>{ + \fill[color=red!20] (7,0) rectangle (8,8); +} +\uncover<11->{ + \fill[color=red!20] (7,0) rectangle (8,1); + \node[color=gray] at (7.5,0.5) {$0$}; + \draw[color=gray] (7.05,0.05) rectangle (7.95,0.95); +} + +\draw (0,0) rectangle (8,8); +\node at (-0.1,4) [left] {$A=$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} \end{frame} +\egroup |