Update parts to newest handwritten Skriptum
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LinAlg2.tex
84
LinAlg2.tex
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@ -3619,11 +3619,13 @@ $ \implies q(\tilde x_1, \tilde x_2) = \lambda_1 \tilde x_1^2 + \lambda_2 \tilde
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\begin{lemma}
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Sei $\operatorname{char}(\K) \neq 2$. Dann entsprechen die quadratischen Formen und symmetrischen
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Bilinearformen eineindeutig.
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Bilinearformen einander eineindeutig.
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\begin{proof}
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$\rho$ quadratische Form $\implies \sigma(v, w) = \rho(u + v) - \rho(u) - \rho(v)$ ist symmetrische
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Bilinearform. \\
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Sei $\sigma$ symmetrische Bilinearform, $\rho(v) := \frac 12 \sigma(v, v)$.
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Sei umgekehrt $\sigma$ symmetrische Bilinearform,
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$\rho(v) := \underset{\mathclap{\substack{\rotatebox{90}{$\to$}\\\operatorname{char}(\K) \neq 2}}}
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{\frac 12} \sigma(v, v)$.
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\begin{align*}
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\rho(\lambda v) = \frac 12 \sigma(\lambda v, \lambda v) & = \lambda^2 \frac 12 \sigma(v, v) =
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\lambda^2 \rho(v) \implies \text{a)} \\
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@ -3639,7 +3641,7 @@ $ \implies q(\tilde x_1, \tilde x_2) = \lambda_1 \tilde x_1^2 + \lambda_2 \tilde
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\begin{defin}
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\label{theo:3.5.6}
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$V \C$-VR. $\rho: V \to \R$ heißt \underline{hermitesche Form} wenn $\forall u, v \in V, \lambda \in \C$:
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Sei $V\, \C$-VR. $\rho: V \to \R$ heißt \underline{hermitesche Form} wenn $\forall u, v \in V, \lambda \in \C$:
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\begin{enumerate}[label=\alph*)]
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\item $\rho(\lambda v) = \abs{\lambda}^2 \rho(v)$
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\item $\rho(u+v) + \rho(u -v) = 2(\rho(u) + \rho(v))$
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@ -3686,17 +3688,20 @@ $\sigma$ heißt Polarform von $\rho$
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\section[Die Singulärwertzerlegung und die Pseudoinverse]{Die Singulärwertzerlegung und die \\Pseudoinverse}
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Wollen Normalform für $\alpha \in \homk(V, W)$ mit $V, W$ euklidisch/unitär herleiten.
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Polarzerlegung: $W = V \implies \exists$ ONBs $B, B'$ von $V$ mit
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Wir wollen nun für zwei euklidische Vektorräume $V, W$ eine geeignete Normalform bezüglich Orthonormalbasen
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herleiten. Polarzerlegung besagt für $\alpha \in \Hom(V, V)$, dass Orthonormalbasen $B, B'$ von $V$ existieren
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mit
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\[
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{}_B M(\alpha)_B = \begin{pmatrix}s_1 \\
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{}_B M(\alpha)_B = \begin{pmatrix}
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s_1 \\
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& \ddots \\
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& & s_r \\
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& & & 0 \\
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& & & & \ddots \\
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& & & & & 0\end{pmatrix}, s_1, \dots, s_n > 0
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& & & & & 0
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\end{pmatrix}, s_1, \dots, s_n > 0
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\]
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Das gilt für $V, W$ allgemein.
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Das heißt $\alpha$ lässt sich aus orthogonalen Endomorphismen und Skalierung zusammensetzen.
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\begin{satz}[Singulärwertzerlegung]
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Sei $A \in \R^{m \times n} / \C^{m \times n}$. Dann gibt es orthogonale/unitäre Matrizen $U, V$ sowie
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@ -3707,21 +3712,23 @@ Das gilt für $V, W$ allgemein.
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& & s_r \\
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& & & 0 \\
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& & & & \ddots \\
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& & & & & 0\end{pmatrix}}_{\K^{m \times m}} \underbrace{V}_{\K^{m \times n}}
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& & & & & 0\end{pmatrix}}_{\K^{m \times n}} \underbrace{V}_{\K^{n \times n}}
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\]
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$s_1, \dots, s_r$ heißen positive \underline{Singulärwerte} von $A$.
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$s_1, \dots, s_r$ heißen \underline{Singulärwerte} von $A$.
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\begin{proof}
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\begin{itemize}
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\item $A^* A \in \K^{\nxn}$ selbstadjungiert und positiv semi-definit.
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Eigenwerte $\lambda_1, \dots, \lambda_n \in [0, \infty)$, ONB $b_1, \dots, b_n$ aus Eigenvektoren.
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Sei $\lambda_1, \dots, \lambda_r \in (0, \infty), \lambda_{r+1} = \dots = \lambda_n = 0$
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$s_i := \sqrt{\lambda_i}, i\in [n]$
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\item $\overbrace{\frac 1{s_1} A b_1}^{b_1'}, \dots, \overbrace{\frac 1{s_r} A b_r}^{b_r'}$ ist
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Orthonormalsystem in $\K^m$.
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\[
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\overline{\inner{Ab_i}{Ab_j}}_{\K^m} = \overline{b_i}^T A^* A b_j = \lambda_j \overline{b_i}^T b_j
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= \lambda_j \overline{\inner{b_i}{b_j}}_{\K^n} = \lambda_j \delta_{ij} \in \R
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\]
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\item Es gilt, dass $\overbrace{\frac 1{s_1} A b_1}^{b_1'}, \dots, \overbrace{\frac 1{s_r} A b_r}^{b_r'}$
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Orthonormalsystem in $\K^m$ ist.
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\begin{align*}
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\overline{\inner{Ab_i}{Ab_j}}_{\K^m} & = \overline{b_i^T A^T \overline A \,\overline{b_j}}
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= \overline{b_i}^T A^* A b_j = \lambda_j \overline{b_i}^T b_j \\
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& = \lambda_j \overline{\inner{b_i}{b_j}}_{\K^n}
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= \lambda_j \delta_{ij} \in \R
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\end{align*}
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\item Ergänze $b_1', \dots, b_r'$ zu Orthonormalbasis $b_1', \dots, b_r', \dots, b_m'$ von $\K^m$. \\
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Sei $\varphi_A: x \mapsto A\cdot x \implies {}_{B'} M(\varphi_A)_B = \left( \begin{smallmatrix}
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s_1 \\
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@ -3736,23 +3743,42 @@ Das gilt für $V, W$ allgemein.
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\end{proof}
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\end{satz}
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Mittels der Singulärwertzerlegung können wir für jede Matrix (bzw. lineare Abbildung) eine verallgemeinerte
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Inverse berechnen.
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Sei ${}_B M(\alpha)_B = \begin{pmatrix}s_1 \\
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& \ddots \\
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& & s_r \\
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& & & 0 \\
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& & & & \ddots \\
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& & & & & 0\end{pmatrix}$ \\
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$\implies \ker(\alpha) = \langle b_{r+1}, \dots, b_n \rangle_V, \im(\alpha) = \langle b'_1, \dots b_r' \rangle_W,
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\ker(\alpha)^\bot = \langle b_1, \dots, b_r \rangle_V$
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\begin{align*}
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\alpha: & V \to
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& & \ker(\alpha)^{\bot} \overset{\beta}{\to}
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& & \im(\alpha) \to
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& & W \\
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& \sum_{i=1}^n \lambda_i b_i \mapsto
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& & \sum_{i=1}^r \lambda_i b_i \mapsto
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& & \sum_{i=1}^r s_i \lambda_i b_i' \mapsto
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& & \sum_{i=1}^r s_i \lambda_i b_i' \\
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& \sum_{i=1}^r \frac{\mu_i}{s_i} b_i \mapsfrom
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& & \sum_{i=1}^r \frac{1}{s_i} \mu_i b_i \underset{\beta^{-1}}{\mapsfrom}
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& & \sum_{i=1}^r \mu_i b_i' \mapsfrom
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& & \sum_{i=1}^m \mu_i b_i' = w \cdot \alpha^{+}
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\alpha: V & \to
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\ker(\alpha)^{\bot} & & \overset{\beta}{\to}
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\im(\alpha) & & \to
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W \\
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\sum_{i=1}^n \lambda_i b_i & \mapsto
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\sum_{i=1}^r \lambda_i b_i & & \mapsto
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\sum_{i=1}^r s_i \lambda_i b_i' & & \mapsto
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\sum_{i=1}^r s_i \lambda_i b_i' \\
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\left(\begin{smallmatrix}x_1 \\ \vdots \\ x_n\end{smallmatrix}\right) & \mapsto
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\left(\begin{smallmatrix}x_1 \\ \vdots \\ x_r\end{smallmatrix}\right) & & \mapsto
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\left(\begin{smallmatrix}s_1 x_1 \\ \vdots \\ s_r x_r\end{smallmatrix}\right) & & \mapsto
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\left.\left(\begin{smallmatrix}
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s_1 x_1 \\ \vdots \\ s_r x_r \\ 0 \\ \vdots \\ 0
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\end{smallmatrix}\right)\right\} m \\
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\sum_{i=1}^r \frac{\mu_i}{s_i} b_i & \mapsfrom
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\sum_{i=1}^r \frac{1}{s_i} \mu_i b_i & & \underset{\beta^{-1}}{\mapsfrom}
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\sum_{i=1}^r \mu_i b_i' & & \mapsfrom
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\sum_{i=1}^m \mu_i b_i' = w \cdot \alpha^{+}
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\end{align*}
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\subsubsection{Bemerkung:}
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$\alpha$ invertierbar, $V=W \implies n = m = r \implies \alpha^+ = \alpha^{-1}$
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$\alpha$ invertierbar, $V=W \implies n = m = r \implies \alpha^+ = \alpha^{-1}$ \\
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Wir haben eine echte Verallgemeinerung.
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\begin{defin}
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\leavevmode
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