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\draw (0.1, 1) -- (-0.1, 1); + \node at (-0.2, 1) {$1$}; + \draw[red] (0,0) circle (1); + \draw[blue, rotate around={45:(0,0)}] (-1/1.41,-1/1.41) rectangle (1/1.41, 1/1.41); + \draw[green, fill] (1,0) circle (0.02); + \draw[green, fill] (0,1) circle (0.02); + \draw[green, fill] (-1,0) circle (0.02); + \draw[green, fill] (0,-1) circle (0.02); + \node at (1.5, -0.2) {$x_1$}; + \node at (-0.2, 1.5) {$x_2$}; + \end{tikzpicture} + \caption{Blau: $\Vert \cdot \Vert_1$, Rot: $\Vert \cdot \Vert_2$, Grün: + $\Vert \cdot \Vert_{\infty}$} + \end{figure} + \item Sei $x \in \R^{n}$ beliebig. Dann ist + \begin{align*} + \Vert x \Vert_1 = \sum_{k=1}^{n} |x_k \cdot 1| + \quad &\stackrel{\text{C.S.U.}}{\le } \quad + \Vert x \Vert_2 \cdot \Vert 1 \Vert_2 = \sqrt{n} \Vert x \Vert_2 \\ + \Vert x \Vert_2 = \sqrt{\sum_{k=1}^{n} |x_k|^2} + &\le \sqrt{\sum_{k=1}^{n} \Vert x_{\infty}\Vert^2} + = \sqrt{n \Vert x\Vert_\infty^2} = \sqrt{n} \Vert x \Vert_{\infty} + .\end{align*} + Außerdem ist + \begin{align*} + &\Vert x \Vert_1 = \sum_{k=1}^{n} |x_k| + = \Vert x \Vert_2 \left( \sum_{k=1}^{n} \underbrace{\frac{|x_k|}{\Vert x \Vert_2}}_{\le 1} \right) + \ge \Vert x \Vert_2 \left( \sum_{k=1}^{n} \frac{|x_k|^{2}}{\Vert x \Vert_2^2} \right) + = \Vert x \Vert_2 \frac{\Vert x \Vert_2^2}{\Vert_x \Vert_2^2} = \Vert x \Vert_2 \\ + &\Vert x \Vert_2 = \sqrt{\sum_{k=1}^{n} |x_k|^2} \ge \sqrt{\Vert x \Vert_{\infty}^2} + = \Vert x \Vert_{\infty} + .\end{align*} + Damit folgt + \begin{align*} + &\Vert x \Vert_2 \le \Vert x \Vert_1 \le \sqrt{n} \Vert x \Vert_2 \\ + &\Vert x \Vert_\infty \le \Vert x \Vert_2 \le \sqrt{n} \Vert x \Vert_\infty + .\end{align*} + Die anderen Kombinationen folgen durch Multiplikation mit $\frac{1}{\sqrt{n}}$. + Für $n \to \infty$ ist $\sqrt{n} \to \infty$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \[ + f(x) = \frac{1 - \cos x}{x} + .\] + \begin{enumerate}[a)] + \item Es ist + \[ + \frac{\d f}{\d x} = \frac{x \sin x - 1 + \cos x}{x^2} + .\] Damit folgt + \[ + k = \frac{x \sin x - 1 + \cos x}{1 - \cos x} + .\] Für $x = \frac{\pi}{2} + 2\pi k$ ist + \[ + k = \frac{x - 1}{1} = x - 1 \xrightarrow{k \to \infty} \infty + ,\] also $f$ schlecht konditioniert. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Die Matrix ist eine $N-1\times N-1$ Matrix, da eine Gleichung für jeden Knoten + bis auf den Referenzknoten existiert. + \item Jeder Knoten hat entweder 2 (Ecken), 3 (Außenkanten) oder 4 (im Inneren) ein oder + ausgehende Kanten. Die mit der Pumpe verbundenen Knoten, haben jeweils eine Kante mehr. + Die zugehörigen Zeilen haben damit immer $1$ $+$ Anzahl der verbundenen Kanten Einträge ungleich + $0$. + \item Die Matrix ist quadratisch und hat vollen Rang und hat damit eine eindeutige Lösung, wenn + $q_p \neq 0$. + \item + \begin{align*} + \begin{pmatrix} + 3 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ + -1 & 2 & 0 & 0 & -1 & 0 & 0 & 0 \\ + 0 & 0 & 3 & -1 & 0 & -1 & 0 & 0 \\ + -1 & 0 & -1 & 4 & -1 & 0 & -1 & 0 \\ + 0 & -1 & 0 & -1 & 3 & 0 & 0 & -1 \\ + 0 & 0 & -1 & 0 & 0 & 2 & -1 & 0 \\ + 0 & 0 & 0 & -1 & 0 & -1 & 3 & -1 \\ + 0 & 0 & 0 & 0 & -1 & 0 & -1 & 2 + \end{pmatrix} + \begin{pmatrix} p_1 \\ p_2 \\ p_3 \\ p_4 \\ p_5 \\ p_6 \\ p_7 \\ p_8 \end{pmatrix} + = + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ q_p \end{pmatrix} + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Sei $x \in \R^{n} \setminus \{0\}$. Dann ist + \begin{align*} + (A_s x, x)_2 &= \left( \frac{1}{2}Ax + \frac{1}{2}A^{T}x, x \right)_2 + = \frac{1}{2} (Ax, x) + \frac{1}{2} (A^{T}x, x) + = \frac{1}{2} x^{T}A^{T}x + \frac{1}{2} x^{T}Ax \\ + &= \frac{1}{2} x^{T} A^{T} x + \frac{1}{2} \left( x^{T} (Ax) \right)^{T} + = \frac{1}{2} x^{T}A^{T}x + \frac{1}{2} x^{T}A^{T}x + = x^{T}A^{T}x + = (Ax, x)_2 + .\end{align*} + Damit folgt die Behauptung. + \item Sei $X \subseteq \{1, 2, \ldots, n\} $ beliebig und $A_X$ nicht positiv definit. Dann + ex. ein $\widetilde{x} \in \R^{|x|} \setminus \{0\} $ mit $A_X x \le 0$. Dann + ergänze $\widetilde{x}$ zu $x \in \R^{n}$ mit + \[ + x_i := \begin{cases} + x_i & i \in X \\ + 0 & \text{sonst} + \end{cases} + .\] Dann ist $x^{T}Ax = \widetilde{x}^{T}A_X\widetilde{x} \le 0$, also ist $A$ nicht positiv + definit. + \item Mit (a) folgt: $A$ g.d. positiv definit, wenn + \[ + A_S = \begin{pmatrix} 2 & - \frac{\alpha}{2} \\ - \frac{\alpha}{2} & 2 \end{pmatrix} + .\] positiv definit ist. Dies ist mit dem Hauptminorenkriterium für symmetrische + Matrizen g.d der Fall, wenn + \begin{align*} + &\begin{gmatrix}[v] + 2 & - \frac{\alpha}{2} \\ - \frac{\alpha}{2} & 2 + \end{gmatrix} + = 4 - \frac{\alpha^2}{4} > 0 + \\ + \iff &16 - \alpha^2 > 0 \\ + \iff &|\alpha| < 4 + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\end{document}