From 5ff87c07aa8c65116fd07707b718f467bc50920b Mon Sep 17 00:00:00 2001
From: Dominic Zimmer <s8dozimm@stud.uni-saarland.de>
Date: Tue, 13 Jun 2023 09:56:01 +0200
Subject: [PATCH] Update

---
 jupyter/The Art of Memory Loss.html  | 412 ++++++++++++++++++---------
 jupyter/The Art of Memory Loss.ipynb | 240 +++++-----------
 2 files changed, 345 insertions(+), 307 deletions(-)

diff --git a/jupyter/The Art of Memory Loss.html b/jupyter/The Art of Memory Loss.html
index 85345fc..0a151ee 100644
--- a/jupyter/The Art of Memory Loss.html	
+++ b/jupyter/The Art of Memory Loss.html	
@@ -15534,9 +15534,9 @@ $$
 </div>
 <div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
 </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
-<p>Da die Übergangswahrscheinlichkeiten zum Zeitpunkt $t$ nur von $X_t$ abhängen, schreiben wir kurzerhand</p>
+<p>Da die Übergangswahrscheinlichkeiten von Zeitpunkt $t$ zu Zeitpunkt $t+1$ nur von $X_t$ abhängen, schreiben wir kurzerhand</p>
 $$
-P(X_t = R | X_{t-1} = L) = P(L \to R) = m_{L R}
+P(X_{t+1} = R | X_{t} = L) = P(L \to R) = m_{L R}
 $$
 </div>
 </div>
@@ -15568,7 +15568,7 @@ $$
 </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
 <p>Wir schreiben die Wahrscheinlichkeiten, dass der Frosch zum Zeitpunkt $t$ in $L$ oder $R$ ist, als Vektoren.</p>
 $$
-X_0 = \begin{pmatrix}1 \\ 0\end{pmatrix}
+X_0 = \begin{pmatrix}1 &amp; 0\end{pmatrix}
 $$
 </div>
 </div>
@@ -15582,165 +15582,240 @@ $$
 </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
 <p>Ein Zeitschritt $t=0 \to t=1$ verändert die Verteilung gemäß $M$:</p>
 $$
-X_0 = \begin{pmatrix}1 \\ 0\end{pmatrix} \to \begin{pmatrix}1-p \\ p \end{pmatrix} = X_1
+X_0 = \begin{pmatrix}1 &amp; 0\end{pmatrix} \to \begin{pmatrix}1-p &amp; p \end{pmatrix} = X_1
 $$
 </div>
 </div>
 </div>
-</div><div id="cell-id=e9f7e2f6" class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs  ">
-<div class="jp-Cell-inputWrapper">
-<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
-<div class="jp-InputArea jp-Cell-inputArea">
-<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[11]:</div>
-<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
-     <div class="CodeMirror cm-s-jupyter">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span> <span class="k">as</span> <span class="n">S</span><span class="p">,</span> <span class="n">Matrix</span> <span class="k">as</span> <span class="n">Mat</span>
-<span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="s2">&quot;p q&quot;</span><span class="p">)</span>
-<span class="n">M</span> <span class="o">=</span> <span class="n">Mat</span><span class="p">([[</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="p">],</span> <span class="p">[</span><span class="n">q</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="p">]])</span>
-</pre></div>
-
-     </div>
-</div>
-</div>
-</div>
-
-</div>
-<div id="cell-id=0d56b780" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div id="cell-id=713b6dda" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
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 </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
-<h1 id="Matrixmultiplikation">Matrixmultiplikation<a class="anchor-link" href="#Matrixmultiplikation">&#182;</a></h1>
-</div>
-</div>
-</div>
-</div><div id="cell-id=c8a105d0" class="jp-Cell jp-CodeCell jp-Notebook-cell   ">
-<div class="jp-Cell-inputWrapper">
-<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
-</div>
-<div class="jp-InputArea jp-Cell-inputArea">
-<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[19]:</div>
-<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
-     <div class="CodeMirror cm-s-jupyter">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="kn">import</span> <span class="n">matrix_power</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">simplify</span>
-
-<span class="n">x_0</span> <span class="o">=</span> <span class="n">Mat</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="p">[</span> <span class="p">(</span><span class="n">x_0</span> <span class="o">@</span> <span class="p">(</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">i</span><span class="p">)))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
-</pre></div>
-
-     </div>
-</div>
-</div>
-</div>
-
-<div class="jp-Cell-outputWrapper">
-<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
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-<div class="jp-OutputArea-child jp-OutputArea-executeResult">
-    
-    <div class="jp-OutputPrompt jp-OutputArea-prompt">Out[19]:</div>
-
-
-
-
-<div class="jp-RenderedText jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/plain">
-<pre>[Matrix([[1, 0]]),
- Matrix([[0.4, 0.6]]),
- Matrix([[0.28, 0.72]]),
- Matrix([[0.256, 0.744]]),
- Matrix([[0.2512, 0.7488]]),
- Matrix([[0.25024, 0.74976]]),
- Matrix([[0.250048, 0.749952]]),
- Matrix([[0.2500096, 0.7499904]]),
- Matrix([[0.25000192, 0.74999808]]),
- Matrix([[0.250000384, 0.749999616000001]])]</pre>
-</div>
-
-</div>
-
-</div>
-
-</div>
-
-</div>
-<div id="cell-id=5db5186d" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
-<div class="jp-Cell-inputWrapper">
-<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
-</div>
-<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
-</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
-<p>Den Folgezustand von $X = \begin{pmatrix}L &amp; R\end{pmatrix}$ mit Übergangsmatrix $M$ berechnen wir als</p>
+<p>Aus einem gegebenen</p>
+$$
+X_0 = \begin{pmatrix}1 &amp; 0 \end{pmatrix}
+$$<p>können wir das <em>Vektor-Matrix-Produkt</em></p>
 $$
 \begin{align*}
-\begin{pmatrix}L' \\ R'\end{pmatrix}^T &amp;= \begin{pmatrix}L &amp; R\end{pmatrix} \cdot M \\
-&amp;= \begin{pmatrix}L &amp; R\end{pmatrix} \cdot \begin{pmatrix} \color{green}{1 - p} &amp; \color{orange}{p} \\ \color{green}{q} &amp; \color{orange}{1-q} \end{pmatrix} \\
-=&amp; \begin{pmatrix}L \cdot \color{green}{(1 - p)} + R \cdot \color{green}{q} \\
-L \cdot \color{orange}{p} + R \cdot \color{orange}{(1-q)} 
-\end{pmatrix}^T
+X_1 &amp;= X_0 \cdot M \\
+&amp;= \begin{pmatrix} 1 &amp; 0 \end{pmatrix}\cdot{}\begin{pmatrix}1-p &amp; p \\q &amp; 1-q\end{pmatrix} = \begin{pmatrix}1-p &amp; p \end{pmatrix}
 \end{align*}
-$$
+$$<p>ausrechnen.</p>
+
 </div>
 </div>
 </div>
 </div>
-<div id="cell-id=782d4f7a" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div id="cell-id=5d09684e" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
 <div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
 </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
 <blockquote>
-<p>Man berechnet die neuen Einträge als &quot;Zeile mal Spalte&quot;</p>
+<p>Matrixmultiplikation rechnet &quot;Zeile mal Spalte&quot;</p>
+</blockquote>
+$$
+\begin{align*}
+&amp; X_0 \cdot M \\
+&amp;= \begin{pmatrix} 1 &amp; 0 \end{pmatrix} \cdot{} \begin{pmatrix} \color{green}{1 - p} &amp; \color{orange}{p} \\ \color{green}{q} &amp; \color{orange}{1-q} \end{pmatrix} \\
+&amp;= \begin{pmatrix} 
+\begin{pmatrix} 1 &amp; 0 \end{pmatrix}\bullet{}\begin{pmatrix} \color{green}{1 - p} \\ \color{green}{q}\end{pmatrix} &amp;
+\begin{pmatrix} 1 &amp; 0 \end{pmatrix}\bullet{}\begin{pmatrix} \color{orange}{p} \\ \color{orange}{1-q} \end{pmatrix}
+\end{pmatrix}\\
+&amp;= \begin{pmatrix} \color{green}{1 - p}&amp; \color{orange}{p}
+\end{pmatrix}
+\end{align*}
+$$<p>Die Einträge von $X_1 = X_0 \cdot M$ errechnen sich genau so, wie die Folgezustände $L'$ und $R'$ aus $L$ und $R$.</p>
+
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=77e76bcd" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<h1 id="Wetterfr%C3%B6sche">Wetterfr&#246;sche<a class="anchor-link" href="#Wetterfr%C3%B6sche">&#182;</a></h1>
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=acbfdea5" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<div class="alert alert-info">
+<p>In Froschland ändert sich das Wetter jeden Tag. Es gibt <strong>Sonnenschein</strong>, <strong>Regen</strong> und <strong>bewölkte</strong> Tage.</p>
+<p>Je nach heutigem Wetter variiert die Wahrscheinlichkeit für das morgige Wetter:</p>
+<ul>
+<li>30% der Tage verschlechtert sich das Wetter $S \to B \to R$</li>
+<li>30% der Tage bleibt es nach wolkigen Tagen wolkig</li>
+<li>10% der Tage schwingt das Wetter <em>extrem</em> um</li>
+<li>40% der Tage regnet es nach Regen weiter</li>
+</ul>
+</div>
+
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=c3ebdfd9" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<div class="alert alert-success">
+<ol>
+<li>Zeichne den Graphen und stelle die Übergangsmatrix $M$ auf</li>
+<li>Implementiere das Froschwetter in python</li>
+<li>Was ist das erwartete Wetter 14 Tage nach Regen, Wolken oder Sonne?</li>
+</ol>
+</div>
+
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=30f295ea" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<p>Die Übergangsmatrix des Froschwetters</p>
+$$
+\begin{align*}
+M = \frac{1}{10} \begin{pmatrix}6 &amp; 3 &amp; 1 \\ 4 &amp; 3 &amp; 3 \\ 1 &amp; 5 &amp; 4 \end{pmatrix}
+= \begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1 \\ 0.4 &amp; 0.3 &amp; 0.3 \\ 0.1 &amp; 0.5 &amp; 0.4 \end{pmatrix}
+\end{align*}
+$$<p>Wenn es heute regnet $X_0 = \begin{pmatrix}1 &amp; 0 &amp; 0 \end{pmatrix}$, ist die Wettervorschau für morgen $X_0 \cdot M$, bzw für in einer Woche $X_0 \cdot M \cdot \dots M = X_0 \cdot M^{14}$.</p>
+
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=e4cb8f44" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<h1 id="Matrixmultiplikation">Matrixmultiplikation<a class="anchor-link" href="#Matrixmultiplikation">&#182;</a></h1>$$
+\begin{align*}
+X_1 &amp;= \begin{pmatrix}1 &amp; 0 &amp; 0\end{pmatrix} \cdot  \begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1 \\ 0.4 &amp; 0.3 &amp; 0.3 \\ 0.1 &amp; 0.5 &amp; 0.4 \end{pmatrix} \\
+&amp;=
+\begin{pmatrix}
+\begin{pmatrix}1 &amp; 0 &amp; 0\end{pmatrix}\bullet \begin{pmatrix} 0.6 \\ 0.4 \\ 0.1 \end{pmatrix} &amp; 
+\begin{pmatrix}1 &amp; 0 &amp; 0\end{pmatrix}\bullet \begin{pmatrix} 0.3 \\ 0.3 \\ 0.5 \end{pmatrix} &amp;
+\begin{pmatrix}1 &amp; 0 &amp; 0\end{pmatrix}\bullet \begin{pmatrix} 0.1 \\  0.3 \\ 0.4 \end{pmatrix}
+\end{pmatrix}
+\end{align*}
+$$<p>Wir können also $X_{t+1} = X_t \cdot{} M$ ausrechnen. Allerdings könnten wir auch $M^{14}$ ausrechnen und einmal $X_0 \cdot M^{14}$ berechnen.</p>
+<p>Wir rechnen $v \cdot M$ als</p>
+<blockquote>
+<p>Zeile mal Spalte, für alle Spalten von $M$</p>
 </blockquote>
 
 </div>
 </div>
 </div>
 </div>
-<div id="cell-id=daddfd1e" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
-<div class="jp-Cell-inputWrapper">
-<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
-</div>
-<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
-</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
-<h2 id="Beispiele">Beispiele<a class="anchor-link" href="#Beispiele">&#182;</a></h2>$$
-\begin{pmatrix}1 &amp; 0\end{pmatrix}\cdot M = \begin{pmatrix}1-p &amp; p\end{pmatrix}
-$$
-</div>
-</div>
-</div>
-</div>
-<div id="cell-id=c1a6597e" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div id="cell-id=da04fda2" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
 <div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
 </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<p>Ein Matrixprodukt $M_1 \cdot M_2$ berechnet sich als</p>
+<blockquote>
+<p>für jede Zeile von $M_1$: Zeile mal Spalte, für alle Spalten von $M_2$</p>
+</blockquote>
 $$
 \begin{align*}
-\begin{pmatrix}1 \\ 1\end{pmatrix}^T\cdot M &amp;= \begin{pmatrix} (1-p) + q \\ p + (1-q)\end{pmatrix}^T \\
-&amp; = \begin{pmatrix} (1-p) + q \\ p +(1-q)\end{pmatrix}^T \\
+M^2 &amp;= \begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1 \\ 0.4 &amp; 0.3 &amp; 0.3 \\ 0.1 &amp; 0.5 &amp; 0.4 \end{pmatrix} \cdot  \begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1 \\ 0.4 &amp; 0.3 &amp; 0.3 \\ 0.1 &amp; 0.5 &amp; 0.4 \end{pmatrix} \\
+&amp;=
+\begin{pmatrix}
+\begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1 \end{pmatrix}\bullet \begin{pmatrix} 0.6 \\ 0.4 \\ 0.1 \end{pmatrix} &amp; 
+\begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1\end{pmatrix}\bullet \begin{pmatrix} 0.3 \\ 0.3 \\ 0.5 \end{pmatrix} &amp;
+\begin{pmatrix}0.6 &amp; 0.3 &amp; 0.1\end{pmatrix}\bullet \begin{pmatrix} 0.1 \\  0.3 \\ 0.4 \end{pmatrix}\\
+\begin{pmatrix}0.4 &amp; 0.3 &amp; 0.3 \end{pmatrix}\bullet \begin{pmatrix} 0.6 \\ 0.4 \\ 0.1 \end{pmatrix} &amp; 
+\begin{pmatrix}0.4 &amp; 0.3 &amp; 0.3\end{pmatrix}\bullet \begin{pmatrix} 0.3 \\ 0.3 \\ 0.5 \end{pmatrix} &amp;
+\begin{pmatrix}0.4 &amp; 0.3 &amp; 0.3\end{pmatrix}\bullet \begin{pmatrix} 0.1 \\  0.3 \\ 0.4 \end{pmatrix}\\
+\begin{pmatrix}0.1 &amp; 0.5 &amp; 0.4\end{pmatrix}\bullet \begin{pmatrix} 0.6 \\ 0.4 \\ 0.1 \end{pmatrix} &amp; 
+\begin{pmatrix}0.1 &amp; 0.5 &amp; 0.4 \end{pmatrix}\bullet \begin{pmatrix} 0.3 \\ 0.3 \\ 0.5 \end{pmatrix} &amp;
+\begin{pmatrix}0.1 &amp; 0.5 &amp; 0.4 \end{pmatrix}\bullet \begin{pmatrix} 0.1 \\  0.3 \\ 0.4 \end{pmatrix}
+\end{pmatrix}
 \end{align*}
 $$
 </div>
 </div>
 </div>
-</div><div id="cell-id=aa2e9aca" class="jp-Cell jp-CodeCell jp-Notebook-cell   ">
+</div>
+<div id="cell-id=72e9e966" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<p>Um $X_{14}$ auszurechnen könnten wir 14 Vektor-Matrix multiplikationen rechnen
+$$
+  X_{14} = X_0 \cdot M \cdot \dots \cdot M
+$$</p>
+<p>Oder stattdessen $M^{14}$ berechnen und mit einer Vektor-Matrix multiplikation
+$$
+X_{14} = X_0 \cdot M^{14}
+$$
+ausrechnen.</p>
+<div class="alert alert-danger">
+Matrix-Matrix multiplikationen sind schlimmer als Vektor-Matrix multiplikationen.
+</div>
+<div class="alert alert-warning">
+Können wir $M^{14}$ irgendwie effizienter ausrechnen als $M \cdot M \cdot \dots \cdot M = M^{14}$
+</div>
+<div class="alert alert-success">
+    $$
+M^{14} = {(M^7)}^2 = {(M \cdot M^6)}^2 = {(M \cdot {(M^3)}^2)}^2 = {(M \cdot {(M \cdot M \cdot M)}^2)}^2
+$$
+</div>
+
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=4bcc0720" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<p>$\leadsto$ Insgesamt fünf M-M Multiplikationen und eine V-M Multiplikation.</p>
+
+</div>
+</div>
+</div>
+</div><div id="cell-id=e9f7e2f6" class="jp-Cell jp-CodeCell jp-Notebook-cell   ">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
 <div class="jp-InputArea jp-Cell-inputArea">
-<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[4]:</div>
+<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[9]:</div>
 <div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
      <div class="CodeMirror cm-s-jupyter">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v1</span> <span class="o">=</span> <span class="n">Mat</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="n">v1</span> <span class="o">*</span> <span class="n">M</span>
+<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#from sympy import symbols as S, Matrix as Mat</span>
+<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">array</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">]])</span>
+<span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="n">x0</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span> <span class="o">@</span> <span class="n">M</span>
 </pre></div>
 
      </div>
@@ -15756,13 +15831,13 @@ $$
 <div class="jp-OutputArea jp-Cell-outputArea">
 <div class="jp-OutputArea-child jp-OutputArea-executeResult">
     
-    <div class="jp-OutputPrompt jp-OutputArea-prompt">Out[4]:</div>
+    <div class="jp-OutputPrompt jp-OutputArea-prompt">Out[9]:</div>
 
 
 
 
-<div class="jp-RenderedLatex jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/latex">
-$\displaystyle \left[\begin{matrix}1 - p &amp; p\end{matrix}\right]$
+<div class="jp-RenderedText jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/plain">
+<pre>array([0.40909225, 0.3484844 , 0.24242335])</pre>
 </div>
 
 </div>
@@ -15771,16 +15846,71 @@ $\displaystyle \left[\begin{matrix}1 - p &amp; p\end{matrix}\right]$
 
 </div>
 
-</div><div id="cell-id=b31deb83" class="jp-Cell jp-CodeCell jp-Notebook-cell   ">
+</div><div id="cell-id=ae437f47" class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs  ">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
 <div class="jp-InputArea jp-Cell-inputArea">
-<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[5]:</div>
+<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[&nbsp;]:</div>
 <div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
      <div class="CodeMirror cm-s-jupyter">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="n">v2</span> <span class="o">=</span> <span class="n">Mat</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="n">v2</span> <span class="o">*</span> <span class="n">M</span>
+<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">linalg</span>
+<span class="n">x0</span> <span class="o">@</span> <span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
+</pre></div>
+
+     </div>
+</div>
+</div>
+</div>
+
+</div>
+<div id="cell-id=90ae2b14" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+<h4 id="Mehr-als-die-halbe-Wahrheit">Mehr als die halbe Wahrheit<a class="anchor-link" href="#Mehr-als-die-halbe-Wahrheit">&#182;</a></h4><p>Sehr brave Markovketten konvergieren mit $t \to \infty$ gegen eine eindeutige stationäre Verteilung $X^*$.
+Mathematisch lässt sich diese Verteilung durch</p>
+$$
+  X^* \cdot M \overset{!}{=} X^*
+$$<p>ausdrücken und explizit als Lösung des Gleichungssystems (denn Matrizen stellen mehrere Gleichungen dar) ausrechnen.
+Um eine stochastische Verteilung zu erhalten, brauchen wir die zusätzliche Gleichung</p>
+$$
+1 = \sum_{i} X^*_i.
+$$<p>Allgemein drückt man Lösungen der oberen Gleichung durch die Form</p>
+$$
+v^* M = \lambda v^*
+$$<p>aus. Hierbei ist $\lambda != 0$ eine Zahl, die man auch <em>Eigenwert</em> nennt. Eine Lösung für die Eigenwertgleichung ist also ein Vektor $v^*$, der sich nach Matrixmultiplikation mit $M$ um den Faktor $\lambda$ streckt, allerdings seine Richtung beibehält. Da in der obigen Gleichung der Streckfaktor $\lambda = 1$ ist, nennen wir $X^*$ <em>Eigenvektor von $M$ zum Eigenwert $1$</em>.</p>
+<p>In python können wir die Eigenwerte und -Vektoren berechnen lassen:</p>
+
+</div>
+</div>
+</div>
+</div><div id="cell-id=aa7a66a1" class="jp-Cell jp-CodeCell jp-Notebook-cell   ">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea">
+<div class="jp-InputPrompt jp-InputArea-prompt">In&nbsp;[13]:</div>
+<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
+     <div class="CodeMirror cm-s-jupyter">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># linalg.eig berechnet Rechts-Eigenvektoren von Matrizen, wir suchen allerdings Links-Eigenvektoren</span>
+<span class="c1"># Um zum richtigen Ergebnis zu kommen, müssen wir M also spiegeln (M.transpose())</span>
+<span class="n">ews</span><span class="p">,</span> <span class="n">evs</span> <span class="o">=</span> <span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">transpose</span><span class="p">())</span> <span class="c1"># berechnet Eigenwerte und Eigenvektoren</span>
+
+<span class="c1"># Wir sehen, dass ews[0] der Eigenwert 1 ist</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">ews</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+
+<span class="c1"># Folglich ist die 0-te Spalte von evs der dazugehörige Eigenvektor</span>
+<span class="n">v</span> <span class="o">=</span> <span class="n">evs</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
+
+<span class="c1"># Wir müssen v auf Länge 1 skalieren, dass v ein stochastischer Vektor ist</span>
+<span class="n">v</span> <span class="o">/=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
+
+<span class="c1"># Wir sehen, dass v und v@M der gleiche Vektor sind.</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">v</span> <span class="o">@</span> <span class="n">M</span><span class="p">)</span>
 </pre></div>
 
      </div>
@@ -15794,15 +15924,17 @@ $\displaystyle \left[\begin{matrix}1 - p &amp; p\end{matrix}\right]$
 
 
 <div class="jp-OutputArea jp-Cell-outputArea">
-<div class="jp-OutputArea-child jp-OutputArea-executeResult">
+<div class="jp-OutputArea-child">
     
-    <div class="jp-OutputPrompt jp-OutputArea-prompt">Out[5]:</div>
+    <div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
 
 
-
-
-<div class="jp-RenderedLatex jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/latex">
-$\displaystyle \left[\begin{matrix}- p + q + 1 &amp; p - q + 1\end{matrix}\right]$
+<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain">
+<pre>0.9999999999999989
+[0.40909091 0.34848485 0.24242424]
+[0.40909091 0.34848485 0.24242424]
+</pre>
+</div>
 </div>
 
 </div>
@@ -15810,9 +15942,7 @@ $\displaystyle \left[\begin{matrix}- p + q + 1 &amp; p - q + 1\end{matrix}\right
 </div>
 
 </div>
-
-</div>
-<div id="cell-id=e9c1e7e7" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div id="cell-id=b7f0b716" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
@@ -15828,7 +15958,7 @@ Wie können wir aus den 7 Fröschen <strong>fair</strong> einen auswählen?</p>
 </div>
 </div>
 </div>
-<div id="cell-id=a372acd0" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div id="cell-id=ddfdfa5e" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
 <div class="jp-Cell-inputWrapper">
 <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
 </div>
@@ -15933,6 +16063,24 @@ Wie können wir aus den 7 Fröschen <strong>fair</strong> einen auswählen?</p>
 <li>jede Zeile summiert sich zu $1$</li>
 </ul>
 
+</div>
+</div>
+</div>
+</div>
+<div id="cell-id=786f76e8" class="jp-Cell jp-MarkdownCell jp-Notebook-cell">
+<div class="jp-Cell-inputWrapper">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput " data-mime-type="text/markdown">
+$$
+\begin{align*}
+X_1 &amp;= X_0 \cdot{} M \\
+X_2 &amp;= X_1 \cdot{} M = (X_0 \cdot M) \cdot{} M = X_0 \cdot M^2 \\
+&amp; \vdots \\
+X_n &amp;= X_0 \cdot \underbrace{M \cdot{} \dots \cdot M}_{\text{$n$ mal}} = X_0 \cdot M^n
+\end{align*}
+$$
 </div>
 </div>
 </div>
diff --git a/jupyter/The Art of Memory Loss.ipynb b/jupyter/The Art of Memory Loss.ipynb
index 3a53bc5..fec11b3 100644
--- a/jupyter/The Art of Memory Loss.ipynb	
+++ b/jupyter/The Art of Memory Loss.ipynb	
@@ -870,7 +870,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7a2f813",
+   "id": "da04fda2",
    "metadata": {
     "hideCode": false,
     "hidePrompt": false,
@@ -909,7 +909,10 @@
     "hidePrompt": false,
     "slideshow": {
      "slide_type": "subslide"
-    }
+    },
+    "tags": [
+     "hide_cell"
+    ]
    },
    "source": [
     "Um $X_{14}$ auszurechnen könnten wir 14 Vektor-Matrix multiplikationen rechnen\n",
@@ -934,13 +937,16 @@
   },
   {
    "cell_type": "markdown",
-   "id": "80c060ae",
+   "id": "f0d467f4",
    "metadata": {
     "hideCode": false,
     "hidePrompt": false,
     "slideshow": {
      "slide_type": "fragment"
-    }
+    },
+    "tags": [
+     "hide_cell"
+    ]
    },
    "source": [
     "<div class=\"alert alert-success\">\n",
@@ -952,7 +958,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a6633c2b",
+   "id": "72e9e966",
    "metadata": {
     "hideCode": false,
     "hidePrompt": false,
@@ -988,7 +994,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "72f34537",
+   "id": "4bcc0720",
    "metadata": {
     "slideshow": {
      "slide_type": "fragment"
@@ -1033,7 +1039,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "73dd4797",
+   "id": "ae437f47",
    "metadata": {
     "slideshow": {
      "slide_type": "fragment"
@@ -1046,171 +1052,74 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "2454075a",
+   "cell_type": "markdown",
+   "id": "d628c382",
    "metadata": {
     "slideshow": {
      "slide_type": "fragment"
     }
    },
-   "outputs": [],
    "source": [
-    "# Alternatively using linalg\n",
-    "# v = linalg.eig(M.transpose())[1][:,0]\n",
-    "# v /= sum(v)\n",
-    "# v @ M"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "0d56b780",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
-   "source": [
-    "# Matrixmultiplikation"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "5db5186d",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
-   "source": [
-    "Den Folgezustand von $X = \\begin{pmatrix}L & R\\end{pmatrix}$ mit Übergangsmatrix $M$ berechnen wir als\n",
+    "#### Mehr als die halbe Wahrheit\n",
+    "\n",
+    "Sehr brave Markovketten konvergieren mit $t \\to \\infty$ gegen eine eindeutige stationäre Verteilung $X^*$.\n",
+    "Mathematisch lässt sich diese Verteilung durch\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\begin{pmatrix}L' \\\\ R'\\end{pmatrix}^T &= \\begin{pmatrix}L & R\\end{pmatrix} \\cdot M \\\\\n",
-    "&= \\begin{pmatrix}L & R\\end{pmatrix} \\cdot \\begin{pmatrix} \\color{green}{1 - p} & \\color{orange}{p} \\\\ \\color{green}{q} & \\color{orange}{1-q} \\end{pmatrix} \\\\\n",
-    "=& \\begin{pmatrix}L \\cdot \\color{green}{(1 - p)} + R \\cdot \\color{green}{q} \\\\\n",
-    "L \\cdot \\color{orange}{p} + R \\cdot \\color{orange}{(1-q)} \n",
-    "\\end{pmatrix}^T\n",
-    "\\end{align*}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "782d4f7a",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "fragment"
-    }
-   },
-   "source": [
-    "> Man berechnet die neuen Einträge als \"Zeile mal Spalte\""
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "daddfd1e",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
-   "source": [
-    "## Beispiele\n",
+    "  X^* \\cdot M \\overset{!}{=} X^*\n",
+    "$$\n",
+    "\n",
+    "ausdrücken und explizit als Lösung des Gleichungssystems (denn Matrizen stellen mehrere Gleichungen dar) ausrechnen.\n",
+    "Um eine stochastische Verteilung zu erhalten, brauchen wir die zusätzliche Gleichung\n",
     "\n",
     "$$\n",
-    "\\begin{pmatrix}1 & 0\\end{pmatrix}\\cdot M = \\begin{pmatrix}1-p & p\\end{pmatrix}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "c1a6597e",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "fragment"
-    }
-   },
-   "source": [
+    "1 = \\sum_{i} X^*_i.\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\begin{pmatrix}1 \\\\ 1\\end{pmatrix}^T\\cdot M &= \\begin{pmatrix} (1-p) + q \\\\ p + (1-q)\\end{pmatrix}^T \\\\\n",
-    "& = \\begin{pmatrix} (1-p) + q \\\\ p +(1-q)\\end{pmatrix}^T \\\\\n",
-    "\\end{align*}\n",
-    "$$"
+    "\n",
+    "Allgemein drückt man Lösungen der oberen Gleichung durch die Form\n",
+    "\n",
+    "$$\n",
+    "v^* M = \\lambda v^*\n",
+    "$$\n",
+    "\n",
+    "aus. Hierbei ist $\\lambda != 0$ eine Zahl, die man auch *Eigenwert* nennt. Eine Lösung für die Eigenwertgleichung ist also ein Vektor $v^*$, der sich nach Matrixmultiplikation mit $M$ um den Faktor $\\lambda$ streckt, allerdings seine Richtung beibehält. Da in der obigen Gleichung der Streckfaktor $\\lambda = 1$ ist, nennen wir $X^*$ *Eigenvektor von $M$ zum Eigenwert $1$*.\n",
+    "\n",
+    "In python können wir die Eigenwerte und -Vektoren berechnen lassen:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
-   "id": "aa2e9aca",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "fragment"
-    }
-   },
+   "execution_count": 13,
+   "id": "947ce4b1",
+   "metadata": {},
    "outputs": [
     {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}1 - p & p\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "Matrix([[1 - p, p]])"
-      ]
-     },
-     "execution_count": 4,
-     "metadata": {},
-     "output_type": "execute_result"
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.9999999999999989\n",
+      "[0.40909091 0.34848485 0.24242424]\n",
+      "[0.40909091 0.34848485 0.24242424]\n"
+     ]
     }
    ],
    "source": [
-    "v1 = Mat([[1, 0]])\n",
-    "v1 * M"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "id": "b31deb83",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "fragment"
-    }
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\left[\\begin{matrix}- p + q + 1 & p - q + 1\\end{matrix}\\right]$"
-      ],
-      "text/plain": [
-       "Matrix([[-p + q + 1, p - q + 1]])"
-      ]
-     },
-     "execution_count": 5,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "v2 = Mat([[1, 1]])\n",
-    "v2 * M"
+    "# linalg.eig berechnet Rechts-Eigenvektoren von Matrizen, wir suchen allerdings Links-Eigenvektoren\n",
+    "# Um zum richtigen Ergebnis zu kommen, müssen wir M also spiegeln (M.transpose())\n",
+    "ews, evs = linalg.eig(M.transpose()) # berechnet Eigenwerte und Eigenvektoren\n",
+    "\n",
+    "# Wir sehen, dass ews[0] der Eigenwert 1 ist\n",
+    "print(ews[0])\n",
+    "\n",
+    "# Folglich ist die 0-te Spalte von evs der dazugehörige Eigenvektor\n",
+    "v = evs[:,0]\n",
+    "\n",
+    "# Wir müssen v auf Länge 1 skalieren, dass v ein stochastischer Vektor ist\n",
+    "v /= sum(v)\n",
+    "\n",
+    "# Wir sehen, dass v und v@M der gleiche Vektor sind.\n",
+    "print(v)\n",
+    "print(v @ M)"
    ]
   },
   {
@@ -1272,7 +1181,9 @@
    },
    "source": [
     "<div class=\"alert alert-warning\">\n",
-    "Von allen Fröschen ist einer am unvertrauenswürdigsten. Die anderen sieben Frösche haben sich geeinigt, dass unter ihnen ausgelost werden soll.    \n",
+    "    \n",
+    "Von allen Fröschen ist einer am unvertrauenswürdigsten. Die anderen sieben Frösche haben sich geeinigt, dass unter *ihnen* ausgelost werden soll.    \n",
+    "    \n",
     "</div>"
    ]
   },
@@ -1405,27 +1316,6 @@
     "- jede Zeile summiert sich zu $1$"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "id": "786f76e8",
-   "metadata": {
-    "hideCode": false,
-    "hidePrompt": false,
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
-   "source": [
-    "$$\n",
-    "\\begin{align*}\n",
-    "X_1 &= X_0 \\cdot{} M \\\\\n",
-    "X_2 &= X_1 \\cdot{} M = (X_0 \\cdot M) \\cdot{} M = X_0 \\cdot M^2 \\\\\n",
-    "& \\vdots \\\\\n",
-    "X_n &= X_0 \\cdot \\underbrace{M \\cdot{} \\dots \\cdot M}_{\\text{$n$ mal}} = X_0 \\cdot M^n\n",
-    "\\end{align*}\n",
-    "$$"
-   ]
-  },
   {
    "cell_type": "code",
    "execution_count": 7,
@@ -1490,7 +1380,7 @@
   }
  ],
  "metadata": {
-  "celltoolbar": "Slideshow",
+  "celltoolbar": "Tags",
   "hide_code_all_hidden": false,
   "kernelspec": {
    "display_name": "Python 3 (ipykernel)",