Fix a few typos
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46
LinAlg2.tex
46
LinAlg2.tex
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@ -376,7 +376,7 @@ $\varphi$ alternierend und $a_i = a_j$ für $i\neq j \implies \varphi(a_1, \dots
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& \underbrace{=}_{\mathclap{\varphi\text{ alternierend}}}
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\sum_{\substack{j_1, \dots, j_n \\
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\text{paarweise verschieden}}}
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{\varphi(b_{j_1}, \dots, v_{j_n})\lambda_{1j_1} \cdots \lambda_{nj_n}} \\
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{\varphi(b_{j_1}, \dots, b_{j_n})\lambda_{1j_1} \cdots \lambda_{nj_n}} \\
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& = \sum_{\pi\in S_n} \varphi(b_{\pi(1)}, \dots, b_{\pi(n)})
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\lambda_{1\pi(1)} \cdots \lambda_{n\pi(n)} \\
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& \underbrace{=}_{\mathclap{\text{Lemma \ref{theo:1.2.3}}}}
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@ -422,9 +422,10 @@ $\varphi$ alternierend und $a_i = a_j$ für $i\neq j \implies \varphi(a_1, \dots
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\varphi(b_i, \dots, b_i, \dots, b_n) & = c\cdot\sum_{\pi\in S_n} \sgn(\pi) \lambda_{i\pi(1)}\cdots\lambda_{i\pi(i)}\cdots\lambda_{n\pi(n)} \\
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& \begin{multlined}=c\cdot \Bigg(\sum_{\pi\in A_n}\sgn(\pi)\lambda_{i\pi(i)}\cdots\lambda_{i\pi(i)}\cdots\lambda_{n\pi(n)} \\
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+\sum_{\pi\in A_n}\underbrace{\sgn(\pi\circ(1i))}_{=-\sgn(\pi)}\lambda_{i\pi(i)}\cdots\lambda_{i\pi(i)}\cdots\lambda_{n\pi(n)}\Bigg)\end{multlined} \\
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& =c\cdot\sum_{\pi\in A_n}(\sgn(\pi)-\sgn(\pi)) \cdot \cdots \\
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& \cdot \cdots \lambda_{i\pi(i)}\cdots\lambda_{i\pi(i)}\cdots\lambda_{n\pi(n)}=0
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\end{align*}
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& \begin{multlined}
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= c\cdot\sum_{\pi\in A_n}(\sgn(\pi)-\sgn(\pi))\\
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\cdot \lambda_{i\pi(i)}\cdots\lambda_{i\pi(i)}\cdots\lambda_{n\pi(n)}=0
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\end{multlined}\end{align*}
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\end{enumerate}
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\end{proof}
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@ -505,7 +506,7 @@ Es gibt also zu jedem $\K$-VR V mit $\dim(V)=n$ eine nicht ausgeartete alternier
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Sei V ein n-dimensionaler \K-Vektorraum. Dann gilt
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\begin{enumerate}[label=\alph*)]
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\item $\alpha\in \homkv \text{ bijektiv } \iff \det(\alpha)\neq0$
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\item $\alpha, \beta \in \homkv \implies \det(\alpha, \beta) = \det(\alpha) \det(\beta)$
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\item $\alpha, \beta \in \homkv \implies \det(\alpha \beta) = \det(\alpha) \det(\beta)$
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\item $\det(\id)=1$
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\item Ist $\alpha\in \homkv$ invertierbar, so gilt $\det(\alpha^{-1})=\det(\alpha)^{-1}$.
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\end{enumerate}
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@ -528,13 +529,13 @@ Es gibt also zu jedem $\K$-VR V mit $\dim(V)=n$ eine nicht ausgeartete alternier
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\begin{align*}
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\det(\alpha\beta) & = \frac{\varphi(\alpha(\beta(a_1)), \dots, \alpha(\beta(a_n)))}
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{\varphi(a_1, \dots, a_n)} \\
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& \begin{multlined}
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=\frac{\varphi(\alpha(\beta(a_1)), \dots, \alpha(\beta(a_n)))}
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{\varphi(\beta(a_1), \dots, \beta(a_n))}\cdot
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\frac{\varphi(\beta(a_1), \dots, \beta(a_n))}
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{\varphi(a_1, \dots, a_n)}
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\end{multlined} \\
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& \underbrace{=}_{\mathclap{\text{Satz \ref{theo:1.3.2}}}}
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& \begin{multlined}
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=\frac{\varphi(\alpha(\beta(a_1)), \dots, \alpha(\beta(a_n)))}
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{\varphi(\beta(a_1), \dots, \beta(a_n))}\cdot
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\frac{\varphi(\beta(a_1), \dots, \beta(a_n))}
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{\varphi(a_1, \dots, a_n)}
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\end{multlined} \\
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& \underbrace{=}_{\mathclap{\text{Satz \ref{theo:1.3.2}}}}
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\det(\alpha)\det(\beta)
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\end{align*}
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\end{enumerate}
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@ -730,9 +731,9 @@ da obige Matrix aus $M_{ij}$ durch Spaltenadditionen hervorgeht.
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\end{satz}
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\begin{proof}
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b) \[\begin{aligned}
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\det(A) & = \det(a_{\_1}, \dots, a_{\_n})= \\
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& =\det(a_{\_1}, \dots, \underbrace{\sum_{l=1}^na_{lj}e_l}_{=a_{\_j}}, \dots, a_{\_n})= \\
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& =\sum_{l=1}^n a_{lj}\det(a_{\_1}, \dots, \underbrace{e_l}_{j}, \dots, a_{\_n}) = \\
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\det(A) & = \det(a_{\_1}, \dots, a_{\_n}) \\
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& =\det(a_{\_1}, \dots, \underbrace{\sum_{l=1}^na_{lj}e_l}_{=a_{\_j}}, \dots, a_{\_n}) \\
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& =\sum_{l=1}^n a_{lj}\det(a_{\_1}, \dots, \underbrace{e_l}_{j}, \dots, a_{\_n}) \\
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& = \sum_{l=1}^n a_{lj}A_{lj}
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\end{aligned}
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\]
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@ -859,7 +860,7 @@ da obige Matrix aus $M_{ij}$ durch Spaltenadditionen hervorgeht.
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\item[$\implies$:] Sei $\alpha$ diagonalisierbar und $B$ eine Basis mit $_B M(\alpha)_B$ Diagonalmatrix.
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Dann gilt
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\begin{align*}
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{}_B M(\alpha)_B & = {}_B M(\id)_C \cdot {}_C M(\alpha)_C \cdot {}_C M(\id)_B \\
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{}_B M(\alpha)_B & = {}_B M(\id)_C \cdot {}_C M(\alpha)_C \cdot {}_C M(\id)_B \\
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& = {{}_C M(\id)_{B}}^{-1} \cdot {}_C M(\alpha)_C \cdot {}_C M(\id)_B
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\end{align*}
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Also ist ${}_C M(\alpha)_C$ diagonalisierbar.
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@ -1292,14 +1293,15 @@ $A = \begin{pmatrix}
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Sei $A\in\K^{\nxn}$ und $\underbrace{\spur(A)}_{\mathclap{\color{red}\text{\dq Spur von $A$ \dq}}}
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:= \sum\limits_{i=1}^n a_{ii}$
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\[\chi_A(\lambda) = (-1)^n\lambda^n + (-1)^{n-1} \spur(A) \lambda^{n-1} + \cdots + \det(A)\]
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\end{lemma}
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\begin{proof}
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$\chi_A(\lambda) = \sum\limits_{\pi \in S_n} \sgn(\pi) \prod\limits_{i=1}^n \tilde{a}_{i\pi(i)}$ mit
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$\tilde{a}_{ij} = \begin{cases} a_{ij} & i\neq j \\ a_{ij} - \lambda & i=j\end{cases}$. \\
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\[
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\chi_A(\lambda) = \sum\limits_{\pi \in S_n} \sgn(\pi) \prod\limits_{i=1}^n \tilde{a}_{i\pi(i)} \text{ mit }
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\tilde{a}_{ij} = \begin{cases} a_{ij} & i\neq j \\ a_{ij} - \lambda & i=j\end{cases}
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\]
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Wenn $\pi\neq \id$ gilt $\deg\left(\prod\limits_{i=1}^n \tilde{a}_{i\pi(i)}\right)\le n-2$,
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da mindestens zwei Elemente vertauscht werden. Die Koeffizienten von Grad $n, n-1$ kann man also aus
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$\prod\limits_{i=1}^n \tilde{a}_{ii} = \prod\limits_{i=1}^n (\tilde{a}_{ii} - \lambda)$ ablesen.
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$\prod\limits_{i=1}^n \tilde{a}_{ii} = \prod\limits_{i=1}^n (a_{ii} - \lambda)$ ablesen.
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Daraus folgt die Behauptung für die höchsten beiden Koeffizienten.
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Weiters gilt $\chi_A(0)=\det(A)$, was die Aussage für den konstanten Koeffizienten zeigt.
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\end{proof}
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@ -3973,11 +3975,11 @@ Wir haben eine echte Verallgemeinerung.
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$\ker(\alpha) \subseteq \ker(\alpha^\dagger \circ \alpha) \subseteq
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\ker(\alpha \circ \alpha^\dagger \circ \alpha)
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= \ker(\alpha) \implies \ker(\alpha) = \ker(\alpha^\dagger \circ \alpha)$
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$\im(\alpha) \subseteq \im(\alpha \circ \alpha^\dagger) \subseteq
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$\im(\alpha) \supseteq \im(\alpha \circ \alpha^\dagger) \supseteq
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\im(\alpha \circ \alpha^\dagger \circ \alpha) = \im(\alpha) \implies
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\im(\alpha) = \im(\alpha \circ \alpha^\dagger)$
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\item $\nu := \alpha^\dagger \circ \alpha$ erfüllt $\nu \circ \nu$ und ist selbstadjungiert
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? für $\nu' := \alpha \circ \alpha^\dagger$ \\
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für $\nu' := \alpha \circ \alpha^\dagger$ \\
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$\implies \underbrace{\ker(\nu)}_{=\ker(\alpha)} \bot \im(\nu)$
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[Sei $v\in \ker(\nu), w = \nu(v) \in \im(\nu) \implies \inner vw = \inner{\nu(v)}{w} = 0$] \\
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$\implies$
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