From f297ac1a4e95db255ff6c8e892904c1743ed57e5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 3 Mar 2021 22:51:16 +0100 Subject: add new slides --- vorlesungen/slides/5/zerlegung.tex | 73 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 73 insertions(+) create mode 100644 vorlesungen/slides/5/zerlegung.tex (limited to 'vorlesungen/slides/5/zerlegung.tex') diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex new file mode 100644 index 0000000..9c20c60 --- /dev/null +++ b/vorlesungen/slides/5/zerlegung.tex @@ -0,0 +1,73 @@ +% +% zerlegung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Zerlegung in Eigenräume} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.38] +\fill[color=blue!20] (0,11) rectangle (4,15); +\fill[color=red!20] (4,0) rectangle (15,11); +\fill[color=red!40] (9,0) rectangle (15,6); +\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11); +\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6); +\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3); +\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); +\draw[line width=0.1pt] (9,0) -- (9,15); +\draw[line width=0.1pt] (12,0) -- (12,15); +\draw (0,0) rectangle (15,15); +\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) {$f_{|\mathcal{E}_{\lambda_1}}$}; +\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$}; +\node at (10.5,4.5) {$(f_2)_{|\mathcal{E}_{\lambda_3}}$}; +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Iteration} +$\Lambda=\{\lambda_1,\dots,\lambda_s\}$ Eigenwerte +\begin{align*} +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] +f_1 &= f_{|V_1} +\\[10pt] +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] +f_1 &= f_{|V_1} +\\ +&\phantom{0}\vdots +\end{align*} +$\Rightarrow$ $f$ hat {\color{blue}Blockdiagonalform} für die Zerlegung +\begin{align*} +V&=\bigoplus_{\lambda\in\Lambda} \mathcal{E}_{\lambda} +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup -- cgit v1.2.1 From 38f326c35243942285c43310751025d2c16801dd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 4 Mar 2021 20:30:33 +0100 Subject: Spektraltheorie 1. Teil --- vorlesungen/slides/5/zerlegung.tex | 44 ++++++++++++++++++++++++++++++++------ 1 file changed, 38 insertions(+), 6 deletions(-) (limited to 'vorlesungen/slides/5/zerlegung.tex') 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} -- cgit v1.2.1