diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/02_msespektral/slides.tex | 11 | ||||
| -rw-r--r-- | vorlesungen/slides/5/Makefile.inc | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/5/bloecke.tex | 141 | ||||
| -rw-r--r-- | vorlesungen/slides/5/chapter.tex | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/5/charpoly.tex | 26 | ||||
| -rw-r--r-- | vorlesungen/slides/5/eigenraeume.tex | 14 | ||||
| -rw-r--r-- | vorlesungen/slides/5/folgerungen.tex | 62 | ||||
| -rw-r--r-- | vorlesungen/slides/5/injektiv.tex | 74 | ||||
| -rw-r--r-- | vorlesungen/slides/5/jordan.tex | 130 | ||||
| -rw-r--r-- | vorlesungen/slides/5/kernbild.tex | 28 | ||||
| -rw-r--r-- | vorlesungen/slides/5/ketten.tex | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/5/motivation.tex | 12 | ||||
| -rw-r--r-- | vorlesungen/slides/5/normalnilp.tex | 229 | ||||
| -rw-r--r-- | vorlesungen/slides/5/zerlegung.tex | 44 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 28 |
15 files changed, 756 insertions, 52 deletions
diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex index dc34236..9be6ce1 100644 --- a/vorlesungen/02_msespektral/slides.tex +++ b/vorlesungen/02_msespektral/slides.tex @@ -18,18 +18,23 @@ \folie{5/ketten.tex} \folie{5/dimension.tex} \folie{5/folgerungen.tex} +\folie{5/injektiv.tex} \folie{5/nilpotent.tex} -% XXX \folie{5/eigenraeume.tex} +\folie{5/eigenraeume.tex} +\folie{5/zerlegung.tex} +\folie{5/normalnilp.tex} +\folie{5/bloecke.tex} % Jordan Normalform \section{Jordan-Normalform} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} -% XXX \folie{5/jordannormalform.tex} +\folie{5/jordanblock.tex} +\folie{5/jordan.tex} % XXX \folie{5/minimalpolynom.tex} % XXX \folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} -% XXX \folie{5/cayleyhamilton.tex} +\folie{5/cayleyhamilton.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 4bed49b..872798e 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -16,6 +16,9 @@ chapter5 = \ ../slides/5/eigenraeume.tex \ ../slides/5/zerlegung.tex \ ../slides/5/normalnilp.tex \ + ../slides/5/bloecke.tex \ + ../slides/5/jordanblock.tex \ ../slides/5/jordan.tex \ + ../slides/5/cayleyhamilton.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/bloecke.tex b/vorlesungen/slides/5/bloecke.tex new file mode 100644 index 0000000..974f238 --- /dev/null +++ b/vorlesungen/slides/5/bloecke.tex @@ -0,0 +1,141 @@ +% +% bloecke.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\sx{1} +\def\sy{0.1} +\def\block#1#2{ + \fill[color=red] ({#1},{-#1}) rectangle ({#1+#2},{-#1-#2}); +} +\def\kreuz#1{ + \draw[color=white,line width=0.1pt] (0,{-#1})--(60,{-#1}); + \draw[color=white,line width=0.1pt] (#1,0)--(#1,-60); +} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Blockgrössen aus $\dim\mathcal{K}^k(A)$ ablesen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.56\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (A) at ({1*\sx},{20*\sy}); +\coordinate (B) at ({2*\sx},{(20+15)*\sy}); +\coordinate (C) at ({3*\sx},{(20+15+10)*\sy}); +\coordinate (D) at ({4*\sx},{(20+15+10+8)*\sy}); +\coordinate (E) at ({5*\sx},{(20+15+10+8+5)*\sy}); +\coordinate (F) at ({6*\sx},{(20+15+10+8+5+2)*\sy}); +\fill[color=darkgreen!20] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F) + -- ({6*\sx},0) -- cycle; + +\fill[color=darkgreen!40] (0,0) -- ({1*\sx},0) -- (A) -- cycle; +\fill[color=darkgreen!40] (A) -- ({2*\sx},{20*\sy}) -- (B) -- cycle; +\fill[color=darkgreen!40] (B) -- ({3*\sx},{(20+15)*\sy}) -- (C) -- cycle; +\fill[color=darkgreen!40] (C) -- ({4*\sx},{(20+15+10)*\sy}) -- (D) -- cycle; +\fill[color=darkgreen!40] (D) -- ({5*\sx},{(20+15+10+8)*\sy}) -- (E) -- cycle; +\fill[color=darkgreen!40] (E) -- ({6*\sx},{(20+15+10+8+5)*\sy}) -- (F) -- cycle; + +\draw[color=darkgreen,line width=1.4pt] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F); + +\draw[color=gray] (A) -- (0,{20*\sy}); +\draw[color=gray] (B) -- (0,{(20+15)*\sy}); +\draw[color=gray] (C) -- (0,{(20+15+10)*\sy}); +\draw[color=gray] (D) -- (0,{(20+15+10+8)*\sy}); +\draw[color=gray] (E) -- (0,{(20+15+10+8+5)*\sy}); +\draw[color=gray] (F) -- (0,{(20+15+10+8+5+2)*\sy}); + +\node at ({0.5*\sx},{0.5*20*\sy}) + [right] {$d_1 = \dim\mathcal{K}^1(A)-\dim\mathcal{K}^0(A)$}; +\node at ({1.5*\sx},{0.5*(20+20+15)*\sy}) + [right] {$d_2 = \dim\mathcal{K}^2(A)-\dim\mathcal{K}^1(A)$}; +\node at ({2.5*\sx},{0.5*(2*20+2*15+1*10)*\sy}) [right] {$d_3$}; +\node at ({3.5*\sx},{0.5*(2*20+2*15+2*10+8)*\sy}) [right] {$d_4$}; +\node at ({4.5*\sx-0.1},{0.5*(2*20+2*15+2*10+2*8+5)*\sy+0.2}) [below right] {$d_5$}; +\node at ({5.5*\sx},{0.5*(2*20+2*15+2*10+2*8+2*5+2)*\sy+0.1}) [below] {$d_6$}; + +\fill (A) circle[radius=0.08]; +\fill (B) circle[radius=0.08]; +\fill (C) circle[radius=0.08]; +\fill (D) circle[radius=0.08]; +\fill (E) circle[radius=0.08]; +\fill (F) circle[radius=0.08]; + +\draw[->] (-0.1,0) -- ({6*\sx+1},0) coordinate[label={$k$}]; +\draw[->] (0,-0.1) -- (0,6.5) coordinate[label={right:$\dim\mathcal{K}^k(A)$}]; + +\foreach \x in {0,1,...,6}{ + \draw ({\sx*\x},{-0.05}) -- ({\sx*\x},0.05); + \node at ({\sx*\x},{-0.1}) [below] {$\x$}; +} + +\node at (0,{60*\sy}) [left] {\llap{$n$}}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.43\textwidth} +\vspace{-10pt} +\begin{center} +\begin{tabular}{>{$}c<{$}|>{$}r<{$}|>{$}c<{$}|>{$}c<{$}} +k&d_k&\# M_k(\Bbbk)\text{-Blöcke}&\text{Beispiel}\\ +\hline +0& 0& &\\ +1& 20& d_1-d_2&5\\ +2& 15& d_2-d_3&5\\ +3& 10& d_3-d_4&2\\ +4& 8& d_4-d_5&3\\ +5& 5& d_5-d_6&3\\ +6& 2& d_6 &2\\ +\end{tabular} +\end{center} +\vspace{-13pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.05] +\fill[color=gray!40] (0,0) rectangle (60,-60); +\node[color=white] at (30,-30) [scale=6] {$A$}; +\kreuz{5} +\kreuz{15} +\kreuz{21} +\kreuz{33} +\kreuz{48} +\node at (0,-2.5) [left] {$k=1$}; +\node at (60,-2.5) [right] {$5$ Blöcke}; +\node at (0,-10) [left] {$k=2$}; +\node at (60,-10) [right] {$5$ Blöcke}; +\node at (0,-18) [left] {$k=3$}; +\node at (60,-18) [right] {$2$ Blöcke}; +\node at (0,-27) [left] {$k=4$}; +\node at (60,-27) [right] {$3$ Blöcke}; +\node at (0,-40.5) [left] {$k=5$}; +\node at (60,-40.5) [right] {$3$ Blöcke}; +\node at (0,-54) [left] {$k=6$}; +\node at (60,-54) [right] {$2$ Blöcke}; +\block{0}{1} +\block{1}{1} +\block{2}{1} +\block{3}{1} +\block{4}{1} +\block{5}{2} +\block{7}{2} +\block{9}{2} +\block{11}{2} +\block{13}{2} +\block{15}{3} +\block{18}{3} +\block{21}{4} +\block{25}{4} +\block{29}{4} +\block{33}{5} +\block{38}{5} +\block{43}{5} +\block{48}{6} +\block{54}{6} +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 5083abf..7698f01 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -14,4 +14,7 @@ folie{5/nilpotent.tex} folie{5/eigenraeume.tex} folie{5/zerlegung.tex} folie{5/normalnilp.tex} +folie{5/bloecke.tex} +folie{5/jordanblock.tex} folie{5/jordan.tex} +folie{5/cayleyhamilton.tex} diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex index 1211b43..63bfee5 100644 --- a/vorlesungen/slides/5/charpoly.tex +++ b/vorlesungen/slides/5/charpoly.tex @@ -20,6 +20,7 @@ $A-\mu I$ singulär ist: \] $\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$ \end{block} +\uncover<2->{% \begin{block}{Zerlegung in Linearfaktoren} $\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$: \[ @@ -27,33 +28,42 @@ $\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$: = (X-\mu_1)\dots (X-\mu_n) \] -\end{block} +\end{block}} +\uncover<3->{% \begin{block}{Fundamentalsatz der Algebra} Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in Linearfaktoren -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<4->{% \begin{block}{Minimalpolynom} Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen -\end{block} +\end{block}} +\uncover<5->{% \begin{proof}[Beweis] \begin{enumerate} -\item +\item<6-> $m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$ -\item +\item<7-> $A-\lambda I$ ist regulär \end{enumerate} +\uncover<8->{% \begin{align*} &\Rightarrow& m_A(A)&=0 \\ && +\uncover<9->{ (A-\lambda)^{-1}m_A(A) &=0 +} \\ && +\uncover<10->{ \prod_{i\in I}(A-\mu_i)&=0, -\end{align*} +} +\end{align*}} +\uncover<11->{% d.~h.~\( \displaystyle \overline{m}_A(X) @@ -61,8 +71,8 @@ d.~h.~\( \prod_i{i\in I}(X-\mu_i) \in \mathbb{C}[X] -\) -\end{proof} +\)} +\end{proof}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex index 5192cbc..fd4803c 100644 --- a/vorlesungen/slides/5/eigenraeume.tex +++ b/vorlesungen/slides/5/eigenraeume.tex @@ -15,19 +15,25 @@ E_\lambda(f) &= \ker (f-\lambda) \\ +\uncover<2->{ &= \{v\in V\;|\; f(v) = \lambda v\} +} \end{align*} -{\em Eigenraum} von $f$ zum Eigenwert $\lambda$. +\uncover<3->{% +{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.} \end{block} -$E_\lambda(f)\subset V$ ist ein Unterraum +\uncover<4->{% +$E_\lambda(f)\subset V$ ist ein Unterraum} +\uncover<5->{% \begin{block}{Eigenwert} Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$. -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<6->{% \begin{block}{verallgemeinerter Eigenraum} Für $\lambda\in \Bbbk$ heisst \[ @@ -36,7 +42,7 @@ Für $\lambda\in \Bbbk$ heisst \mathcal{K}(f-\lambda) \] verallgemeinerter Eigenraum -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex index a1fa6bf..4a8dbe6 100644 --- a/vorlesungen/slides/5/folgerungen.tex +++ b/vorlesungen/slides/5/folgerungen.tex @@ -3,10 +3,14 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % +\bgroup +\def\sx{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} \begin{frame}[t] \frametitle{Folgerungen} +\vspace{-10pt} \begin{columns}[t] -\begin{column}{0.48\textwidth} +\begin{column}{0.30\textwidth} \begin{block}{Zunahme} Für alle $k<l$ gilt \begin{align*} @@ -23,10 +27,58 @@ Für $k\ge l$ gilt Ausserdem ist $l\le n$ \end{block} \end{column} -\begin{column}{0.48\textwidth} -\begin{proof}[Beweis] -XXX -\end{proof} +\begin{column}{0.66\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\pfad{ + ({0*\sx},6) -- + ({1*\sx},4.5) -- + ({2*\sx},3.5) -- + ({3*\sx},2.9) -- + ({4*\sx},2.6) -- + ({5*\sx},2.4) -- + ({6*\sx},2.4) +} + +\fill[color=orange!20] \pfad -- ({6*\sx},0) -- (0,0) -- cycle; +\fill[color=darkgreen!20] \pfad -- ({6*\sx},6) -- cycle; +\fill[color=orange!40] ({5*\sx},0) rectangle ({6*\sx},2.4); +\fill[color=darkgreen!40] ({5*\sx},6) rectangle ({6*\sx},2.4); + +\draw[color=darkgreen,line width=2pt] ({3*\sx},6) -- ({3*\sx},2.9); +\node[color=darkgreen] at ({3*\sx},4.45) [rotate=90,above] {$\dim\mathcal{K}^k(A)$}; +\draw[color=orange,line width=2pt] ({3*\sx},0) -- ({3*\sx},2.9); +\node[color=orange] at ({3*\sx},1.45) [rotate=90,above] {$\dim\mathcal{J}^k(A)$}; + +\node[color=orange] at ({5.5*\sx},1.2) [rotate=90] {bijektiv}; +\node[color=darkgreen] at ({5.5*\sx},4.2) [rotate=90] {konstant}; + +\fill ({0*\sx},6) circle[radius=0.08]; +\fill ({1*\sx},4.5) circle[radius=0.08]; +\fill ({2*\sx},3.5) circle[radius=0.08]; +\fill ({3*\sx},2.9) circle[radius=0.08]; +\fill ({4*\sx},2.6) circle[radius=0.08]; +\fill ({5*\sx},2.4) circle[radius=0.08]; +\fill ({6*\sx},2.4) circle[radius=0.08]; + +\draw \pfad; + +\draw[->] (-0.1,0) -- ({6*\sx+0.5},0) coordinate[label={$k$}]; +\draw[->] (-0.1,6) -- ({6*\sx+0.5},6); + +\foreach \x in {0,...,6}{ + \draw (\x,-0.05) -- (\x,0.05); +} +\foreach \x in {0,...,3}{ + \node at ({\x*\sx},-0.05) [below] {$\x$}; +} +\node at ({4*\sx},-0.05) [below] {$\dots\mathstrut$}; +\node at ({5*\sx},-0.05) [below] {$l$}; +\node at ({6*\sx},-0.05) [below] {$l+1$}; + +\end{tikzpicture} +\end{center} \end{column} \end{columns} \end{frame} +\egroup diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex index e08cafd..90cbcd6 100644 --- a/vorlesungen/slides/5/injektiv.tex +++ b/vorlesungen/slides/5/injektiv.tex @@ -3,7 +3,79 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % +\bgroup +\def\sx{1.05} \begin{frame}[t] \frametitle{$f$ injektiv auf $\mathcal{J}(f)$} -XXX +\setlength{\abovedisplayskip}{8pt} +\setlength{\belowdisplayskip}{8pt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.58\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=orange!20] + ({0*\sx},-3.0) -- ({1*\sx},-2.0) -- ({2*\sx},-1.5) -- + ({3*\sx},-1.1) -- ({4*\sx},-0.9) -- ({5*\sx},-0.8) -- + ({6*\sx},-0.8) -- + ({6*\sx},0.8) -- ({5*\sx},0.8) -- ({4*\sx},0.9) -- + ({3*\sx},1.1) -- ({2*\sx},1.5) -- ({1*\sx},2.0) -- + ({0*\sx},3.0) -- cycle; +\fill[color=orange!40] (0,-0.8) rectangle ({6*\sx},0.8); + +\foreach \x in {0,...,6}{ + \draw[color=gray,line width=3pt] ({\x*\sx},-3)--({\sx*\x},3); +} +\foreach \x in {0,1,2,3}{ + \node at ({\sx*\x},-3) [below] {$\x$}; +} +\node at ({\sx*5},-3) [below] {$l$}; +\node at ({\sx*6},-3) [below] {$l+1$}; +\draw[->] (-0.1,-3.5) -- ({6*\sx+0.4},-3.5) coordinate[label={below:$k$}]; + +\draw[line width=3pt,color=orange] ({0*\sx},-3.0) -- ({0*\sx},3.0); +\draw[line width=3pt,color=orange] ({1*\sx},-2.0) -- ({1*\sx},2.0); +\draw[line width=3pt,color=orange] ({2*\sx},-1.5) -- ({2*\sx},1.5); +\draw[line width=3pt,color=orange] ({3*\sx},-1.1) -- ({3*\sx},1.1); +\draw[line width=3pt,color=orange] ({4*\sx},-0.9) -- ({4*\sx},0.9); +\draw[line width=3pt,color=orange] ({5*\sx},-0.8) -- ({5*\sx},0.8); +\draw[line width=3pt,color=orange] ({6*\sx},-0.8) -- ({6*\sx},0.8); + +\foreach \x in {0,1,2,3}{ + \node at ({\x*\sx},0) [rotate=90] {$\mathcal{J}^{\x}(A)$}; +} +\node at ({4*\sx},0) {$\cdots$}; +\node at ({5*\sx},0) [rotate=90] {$\mathcal{J}^{l}(A)$}; +\node at ({6*\sx},0) [rotate=90] {$\mathcal{J}^{l+1}(A)$}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.38\textwidth} +\begin{block}{stationär} +$l$ der $k$-Wert, ab dem gilt +\begin{align*} +\mathcal{J}^l(A) &= \mathcal{J}^{l+1}(A) = A\mathcal{J}^l(A) +\end{align*} +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Dimension} +\vspace{-10pt} +\[ +\dim \mathcal{J}^l(A) = \dim\mathcal{J}^{l+1}(A) +\] +\uncover<3->{% +d.~h.~$A$ ist bijektiv als Selbstabbildung von +$\mathcal{J}(A)$} +\uncover<4->{% +\[ +\Downarrow +\] +$A|\mathcal{J}(A)$ ist injektiv} +\end{block}} +\end{column} +\end{columns} \end{frame} +\egroup diff --git a/vorlesungen/slides/5/jordan.tex b/vorlesungen/slides/5/jordan.tex index ad0e31d..e6ece47 100644 --- a/vorlesungen/slides/5/jordan.tex +++ b/vorlesungen/slides/5/jordan.tex @@ -3,8 +3,136 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\L#1{ + \node at ({#1-0.5},{0.5-#1}) {$\lambda$}; +} +\def\E#1{ + \node at ({#1-0.5},{1.5-#1}) {$1$}; +} + \begin{frame}[t] \frametitle{Jordan Normalform} -XXX +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Wahl der Basis} +\begin{enumerate} +\item<2-> Zerlegung in verallgemeinerte Eigenräume +\begin{align*} +V +&= +\mathcal{E}_{{\color{blue}\lambda}}(A) +\oplus +\mathcal{E}_{{\color{darkgreen}\lambda}}(A) +\oplus +\mathcal{E}_{{\color{red}\lambda}}(A) +%\oplus +%\dots +\\ +\llap{$A\mathcal{E}_{{\color{blue}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{blue}\lambda}}(A) +\\ +\llap{$A\mathcal{E}_{{\color{darkgreen}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{darkgreen}\lambda}}(A) +\\ +\llap{$A\mathcal{E}_{{\color{red}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{red}\lambda}}(A), +\dots +\end{align*} +\item<3-> In jedem Eigenraum: Zerlegung in Jordan-Blöcke +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.33] +\fill[color=gray!20] (0,-20) rectangle (20,0); +\node[color=white] at (10,-10) [scale=12] {$A$}; + +\uncover<2->{ + \fill[color=blue!20,opacity=0.5] (0,0) rectangle (8,-8); + \fill[color=darkgreen!20,opacity=0.5] (8,-8) rectangle (15,-15); + \fill[color=red!20,opacity=0.5] (15,-15) rectangle (20,-20); + \fill[color=blue!20] (0,0) rectangle (8,2); + \fill[color=blue!20] (-2,-8) rectangle (0,0); + \fill[color=darkgreen!20] (8,0) rectangle (15,2); + \fill[color=darkgreen!20] (-2,-15) rectangle (0,-8); + \fill[color=red!20] (15,0) rectangle (20,2); + \fill[color=red!20] (-2,-20) rectangle (0,-15); +} + +\uncover<3->{ + \draw[color=gray] (0,0) rectangle (5,-5); + \draw[color=gray] (5,-5) rectangle (8,-8); + \draw[color=gray] (8,-8) rectangle (15,-15); + \draw[color=gray] (15,-15) rectangle (16,-16); + \draw[color=gray] (16,-16) rectangle (17,-17); + \draw[color=gray] (17,-17) rectangle (20,-20); +} + +\uncover<2->{ + \draw[color=gray] (8,0) -- (8,-20); + \draw[color=gray] (15,0) -- (15,-20); + \draw[color=gray] (0,-8) -- (20,-8); + \draw[color=gray] (0,-15) -- (20,-15); + + \node at (0,-4) [above,rotate=90] + {$\mathcal{E}_{{\color{blue}\lambda}}(A)$}; + \node at (4,0) [above] + {$\mathcal{E}_{{\color{blue}\lambda}}(A)$}; + \node at (0,-11.5) [above,rotate=90] + {$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$}; + \node at (11.5,0) [above] + {$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$}; + \node at (0,-18.5) [above,rotate=90] + {$\mathcal{E}_{{\color{red}\lambda}}(A)$}; + \node at (18.5,0) [above] + {$\mathcal{E}_{{\color{red}\lambda}}(A)$}; +} +\uncover<2->{ + {\color{blue} + \foreach \x in {1,...,8}{ \L{\x} } + } + {\color{darkgreen} + \foreach \x in {9,...,15}{ \L{\x} } + } + {\color{red} + \foreach \x in {16,...,20}{ \L{\x} } + } +} + +\uncover<3->{ +\E{2} +\E{3} +\E{4} +\E{5} + +\E{7} +\E{8} + +\E{10} +\E{11} +\E{12} +\E{13} +\E{14} +\E{15} + +\E{19} +\E{20} +} + +\draw (0,-20) rectangle (20,0); +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} \end{frame} + +\egroup diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex index f0bd6fa..3890717 100644 --- a/vorlesungen/slides/5/kernbild.tex +++ b/vorlesungen/slides/5/kernbild.tex @@ -10,29 +10,38 @@ \vspace{-15pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<1->{% \begin{block}{Kern} Lineare Abbildung $f\colon V\to V$ \[ \ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\} \] -\end{block} +\end{block}} +\uncover<3->{% \begin{block}{Kern von $A^k$} \[ \mathcal{K}^k(f) = \operatorname{ker} f^k \] \begin{align*} +\uncover<5->{ \mathcal{K}^k(f) &= \{v\in V\;|\; f^{k}(v)=0\} +} \\ +\uncover<6->{ &\subset \{v\in V\;|\; f^{k+1}(v)=0\} +} \\ +\uncover<7->{ &=\mathcal{K}^{k+1}(f) +} \end{align*} -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% \begin{block}{Bild} Lineare Abbildung $f\colon V\to V$ \[ @@ -42,27 +51,36 @@ Lineare Abbildung $f\colon V\to V$ = \{f(v)\;|\; v\in V\} \] -\end{block} +\end{block}} +\uncover<4->{% \begin{block}{Bild von $A^k$} \[ \mathcal{J}^k(f) = \operatorname{im}f^k \] \begin{align*} +\uncover<8->{ \mathcal{J}^k(f) &= \operatorname{im}f^k = \operatorname{im}(f^{k}\circ f) +} \\ +\uncover<9->{ &= \{f^{k-1} w\;|\; w = f(v)\} +} \\ +\uncover<10->{ &\subset \{f^{k-1} w\;|\; w \in V\} +} \\ -&\mathcal{J}^{k-1}(f) +\uncover<11->{ +&=\mathcal{J}^{k-1}(f) +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex index 759d964..1116a83 100644 --- a/vorlesungen/slides/5/ketten.tex +++ b/vorlesungen/slides/5/ketten.tex @@ -33,6 +33,7 @@ Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt: \] \end{block} \vspace{-20pt} +\uncover<2->{% \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Abildung der Kerne} @@ -74,5 +75,5 @@ f\mathcal{J}(f)&= \mathcal{J}(f) \end{align*} \end{block} \end{column} -\end{columns} +\end{columns}} \end{frame} diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex index 4e8142d..b0a1d82 100644 --- a/vorlesungen/slides/5/motivation.tex +++ b/vorlesungen/slides/5/motivation.tex @@ -13,6 +13,7 @@ Matrix $A$ mit Minimalpolynom $m_A(X)$ vom Grad $s$ \end{block} +\uncover<2->{% \begin{block}{Faktorisieren} Minimalpolynom faktorisieren \[ @@ -20,7 +21,8 @@ m_A(X) = (X-\mu_1)(X-\mu_2)\dots(X-\mu_s) \] -\end{block} +\end{block}} +\uncover<3->{% \begin{block}{Vertauschen} $\sigma\in S_s$ eine Permutation von $1,\dots,s$ ist @@ -39,16 +41,18 @@ m_A(X) \dots (A-\mu_{\sigma(s)}) \end{align*} -\end{block} +\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} +\end{block}} +\uncover<5->{% \begin{block}{Eigenwerte} Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$ gibt mit @@ -57,7 +61,7 @@ gibt mit \Rightarrow Av = {\color{red}\mu} v \] Eigenwertbedingung -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} 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 diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex index 9c20c60..a734d69 100644 --- a/vorlesungen/slides/5/zerlegung.tex +++ b/vorlesungen/slides/5/zerlegung.tex @@ -14,27 +14,48 @@ \begin{column}{0.48\textwidth} \begin{center} \begin{tikzpicture}[>=latex,thick,scale=0.38] +\uncover<2->{ \fill[color=blue!20] (0,11) rectangle (4,15); \fill[color=red!20] (4,0) rectangle (15,11); +} +\uncover<3->{ \fill[color=red!40] (9,0) rectangle (15,6); \fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11); +} +\uncover<4->{ \fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6); \fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3); +} + +\uncover<2->{ \draw[line width=0.1pt] (0,11) -- (15,11); -\draw[line width=0.1pt] (0,6) -- (15,6); -\draw[line width=0.1pt] (0,3) -- (15,3); \draw[line width=0.1pt] (4,0) -- (4,15); +} + +\uncover<3->{ +\draw[line width=0.1pt] (0,6) -- (15,6); \draw[line width=0.1pt] (9,0) -- (9,15); +} + +\uncover<4->{ +\draw[line width=0.1pt] (0,3) -- (15,3); \draw[line width=0.1pt] (12,0) -- (12,15); +} \draw (0,0) rectangle (15,15); +\uncover<2->{ \node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$}; -\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$}; -\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$}; -\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$}; \node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$}; \node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$}; +} +\uncover<3->{ +\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$}; \node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$}; +} +\uncover<4->{ +\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$}; +\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$}; \node at (10.5,4.5) {$(f_2)_{|\mathcal{E}_{\lambda_3}}$}; +} \end{tikzpicture} \end{center} \end{column} @@ -42,30 +63,41 @@ \begin{block}{Iteration} $\Lambda=\{\lambda_1,\dots,\lambda_s\}$ Eigenwerte \begin{align*} +\uncover<2->{ V &= \mathcal{K}(f-\lambda_1) \oplus \raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!20] (0,0) rectangle (1.83,1.1);}}}} \underbrace{\mathcal{J}(f-\lambda_1)}_{\displaystyle=V_1} +} \\[-15pt] +\uncover<2->{ f_1 &= f_{|V_1} +} \\[10pt] +\uncover<3->{ V_1 &= \mathcal{K}(f_1-\lambda_2) \oplus \raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!40] (0,0) rectangle (1.9,1.1);}}}} \underbrace{\mathcal{J}(f_1-\lambda_2)}_{\displaystyle=V_2} +} \\[-15pt] +\uncover<3->{ f_1 &= f_{|V_1} +} \\ +\uncover<4->{ &\phantom{0}\vdots +} \end{align*} +\uncover<5->{% $\Rightarrow$ $f$ hat {\color{blue}Blockdiagonalform} für die Zerlegung \begin{align*} V&=\bigoplus_{\lambda\in\Lambda} \mathcal{E}_{\lambda} -\end{align*} +\end{align*}} \end{block} \end{column} \end{columns} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index c2d361f..2538f29 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -48,31 +48,31 @@ \section{Eigenwertproblem} % XXX Motivation: beliebige Funktionen f(A) berechnen -\folie{5/motivation.tex} -\folie{5/charpoly.tex} +%\folie{5/motivation.tex} +%\folie{5/charpoly.tex} \section{Invariante Unterräume} -\folie{5/kernbild.tex} -\folie{5/ketten.tex} -\folie{5/dimension.tex} -\folie{5/folgerungen.tex} -\folie{5/injektiv.tex} -\folie{5/nilpotent.tex} -\folie{5/eigenraeume.tex} -\folie{5/zerlegung.tex} -\folie{5/normalnilp.tex} +%\folie{5/kernbild.tex} +%\folie{5/ketten.tex} +%\folie{5/dimension.tex} +%\folie{5/folgerungen.tex} +%\folie{5/injektiv.tex} +%\folie{5/nilpotent.tex} +%\folie{5/eigenraeume.tex} +%\folie{5/zerlegung.tex} +%\folie{5/normalnilp.tex} +%\folie{5/bloecke.tex} % Jordan Normalform \section{Jordan-Normalform} +\folie{5/jordanblock.tex} \folie{5/jordan.tex} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} -% XXX \folie{5/jordannormalform.tex} -% XXX \folie{5/minimalpolynom.tex} % XXX \folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} -% XXX \folie{5/cayleyhamilton.tex} +\folie{5/cayleyhamilton.tex} |