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$\chi_{A}^{\text{min}} \in K[t] \setminus \{0\} $ mit $I_A = \left( \chi_{A}^{\text{min}} \right) $. \begin{proof} Es ist $\chi_{A}^{\text{char}} \neq 0$ und $\chi_{A}^{\text{char}} \in I_A$, also @@ -76,7 +77,8 @@ \iff & f(A) = 0 \\ \stackrel{S \neq 0}{\iff} & S f(A) S^{-1} = 0 \\ \iff& f(SAS^{-1}) = 0 \\ - \iff & f(B) = 0 + \iff & f(B) = 0 \\ + \iff & f \in I_B .\end{salign*} Also ist $(\chi_{A}^{\text{min}}) = I_A = I_B = (\chi_{B}^{\text{min}})$ und, wegen $\chi_{A}^{\text{min}}$ und $\chi_{B}^{\text{min}} $ normiert, diff --git a/sose2020/num/uebungen/num5.pdf b/sose2020/num/uebungen/num5.pdf new file mode 100644 index 0000000000000000000000000000000000000000..81b4652784c46da452a8c10c68a8ee1fed730756 GIT binary patch literal 121290 zcmbrlWmFt$vNepmOK^8@+}+(Bf?I;SySuvuO>lPz8r*|Ja0~8Ef6UC8yWVvV>rQ5U z%@6u#*;9M(s%i=)F$o4{Ms_%gsh|JK=Gn`7TtaOO}m@*%SS` zoF9>Fq8&^<`Fcw+0ZyRuvmY&m;+xjQ>Fb@1A){f*J~53~5*1>YA-JWppO1)vewWX9 z-OXvV`BAjFpP-ko$L-vLoLMZ_xQnYtinId8Bt3dCPLe)jJ&{HA<@g-0(az1SZ@oYs 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0$. + Da $A$ symmetrisch bzw. hermitesch, gilt dann: + \[ + \sup_{x \in \mathbb{K}^{n} \setminus \{0\} } \frac{\Vert A x \Vert_2}{\Vert x \Vert_2} + = \max_{1 \le i \le n} | \lambda_i | = \lambda_{\text{max}}(A) + .\] Weiter gilt $\forall x \in \mathbb{K}^{n}$: + \begin{align*} + \frac{(Ax, x)_2}{(x,x)_2} \quad \stackrel{\text{C.S.U.}}{\le} \quad + \frac{\Vert Ax \Vert_2 \Vert x \Vert_2}{\Vert x \Vert_2^2} + = \frac{\Vert Ax \Vert_2}{\Vert x \Vert_2} + .\end{align*} + Damit folgt + \begin{align*} + \sup_{x \in \mathbb{K}^{n} \setminus \{0\} } R_A(x) + \le \sup_{x \in \mathbb{K}^{n} \setminus \{0\} } \frac{\Vert A x \Vert_2}{\Vert x \Vert_2} + = \lambda_{\text{max}}(A) + .\end{align*} + Also ist $\lambda_{\text{max}}(A)$ eine obere Schranke von $R_A(x)$. + + Weiter existiert ein Eigenvektor $v \in \mathbb{K}^{n}$ + zum Eigenwert $\lambda_{\text{max}}(A)$ mit + $Av = \lambda_{\text{max}}(A) v$. Damit folgt + \begin{align*} + R_A(v) = \frac{(Av, v)_2}{(v,v)_2} = \frac{\lambda_{\text{max}}(A)(v,v)_2}{(v,v)_2} + = \lambda_{\text{max}}(A) + .\end{align*} + Also folgt die Behauptung für das Supremum. + + Für das Infimum gilt + \begin{align*} + \inf_{x \in \mathbb{K}^{n} \setminus \{0\} } R_A(x) &= + - \sup_{x \in \mathbb{K}^{n} \setminus \{0\} } - R_A(x) \\ + &\ge - \sup_{x \in \mathbb{K}^{n} \setminus \{0\} } - \frac{\Vert Ax \Vert}{\Vert x \Vert_2} \\ + &= - \max_{1 \le i \le n} - | \lambda_i | \\ + &= \min_{1 \le i \le n} |\lambda_i| \\ + &= \lambda_{\text{min}}(A) + .\end{align*} + Also ist $\lambda_{\text{min}}(A)$ eine untere Schranke von $R_A(x)$. Analog + zu $\lambda_{\text{max}}(A)$ existiert wieder ein Eigenvektor, sodass + die Infimumseigenschaft folgt. + \end{proof} + \item Beh.: + \[ + \text{cond}_2(A) = \Vert A \Vert_2 \Vert A^{-1} \Vert_2 = \frac{\lambda_{\text{max}}(A)}{\lambda_{\text{min}}(A)} + .\] + \begin{proof} + $A$ ist wie in (a) immer noch symmetrisch bzw. hermitesch. Dann gilt nach VL + \[ + \Vert A \Vert_2 = \lambda_{\text{max}}(A) + .\] Außerdem existiert $A^{-1}$, da $A$ positiv definit und symmetrisch bzw. hermitesch und + damit alle Eigenwerte positiv. Weiter ist $A^{-1}$ ebenfalls symmetrisch bzw. hermitesch, + denn $A$ ist symmetrisch bzw. hermitesch und damit + $\overline{A^{-1}}^{T} = \left( \bar{A}^{T} \right)^{-1} = A^{-1}$. + + Weiter gilt $\lambda \neq 0$ Eigenwert von $A$, dann ist $\frac{1}{\lambda}$ Eigenwert + von $A^{-1}$ zu den selben Eigenvektoren, denn + \[ + Av = \lambda v \implies A^{-1} A v = \lambda A^{-1} v \implies v = \lambda A^{-1} v \implies + \frac{1}{\lambda} v = A^{-1} v + .\] Da $A$ positiv definit und symmetrisch bzw. hermitesch, sind alle Eigenwerte positiv und + damit $\lambda_{\text{max}}(A^{-1}) = \frac{1}{\lambda_{\text{min}}(A)}$. Damit folgt + \begin{align*} + \text{cond}_2(A) = \Vert A \Vert_2 \Vert A^{-1} \Vert_2 = \lambda_{\text{max}}(A) + \lambda_{\text{max}}(A^{-1}) = \frac{\lambda_{\text{max}}(A)}{\lambda_{\text{min}}(A)} + .\end{align*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Ansatz 1. Bezeichne + \[ + G := \{ A \in \mathbb{K}^{n \times n} \mid A \text{ untere Dreicksmatrix mit 1-en auf Hauptdiagonale }\} + .\] + \begin{enumerate}[a)] + \item Beh.: $G$ ist Gruppe. + \begin{proof} + \begin{enumerate}[(G1)] + \item Seien $A, B \in G$ mit $A = (a_{ij})_{i,j=1}^{n}$ und $B = (b_{ij})_{i,j=1}^{n}$. + Dann ist $C = AB$ mit $C = (c_{ij})_{i,j=1}^{n}$. Wegen + $A$, $B \in G$ , gilt $a_{ij} = b_{ij} = 0$ für $i < j$ und + $a_{ij} = b_{ij} = 1$ für $i = j$. Damit folgt: + \begin{salign*} + c_{ij} &= \sum_{k=1}^{n} a_{ik} b_{kj} + = \sum_{k=j}^{i} a_{ik}b_{kj} + = \begin{cases} + 0 & i < j \\ + 1 & i = j \\ + \sum_{k=j}^{i} a_{ik}b_{kj} & \text{sonst} + \end{cases} + .\end{salign*} + Also ist $C \in G$. + \item Das neutrale Element ist die Einheitsmatrix $E_n \in \mathbb{K}^{n \times n}$. Diese + ist untere Dreiecksmatrix mit 1-en auf Hauptdiagonale also $E_n \in G$. + \item Sei $A \in G$. Dann ist $\text{det}(A) = 1$, wegen der Dreiecksgestalt und + allen Hauptdiagonalelementen gleich $1$. Also ex. $A^{-1} \in \mathbb{K}^{n \times n}$. + + Zz.: $A^{-1} \in G$. Betrachte die Adjunkte $\tilde{A}$ zu $A$ mit + Einträgen $\tilde{a}_{ij} = (-1)^{i+j} | A_{ji}|$, + wobei $A_{ji}$ die Matrix bezeichnet, die durch Streichen der $j$-ten Zeile + und $i$-ten Spalte in $A$ entsteht. + + Seien $1 \le i,j \le n$. Falls $i = j$. Dann ist + $\tilde{a}_{ii} = (-1)^{2i} | A_{ii}| = |\underbrace{A_{ii}}_{\in G}| = 1$. + Falls $i < j$. Dann ist $A_{ij}$ obere Dreiecksmatrix mit $0$ auf der + Hauptdiagonale, oder eine $4 \times 4$ Blockmatrix, mit zwei Nullblöcken nebeneinander. + Also $|A_{ij}| = 0$ und damit $\tilde{A} \in G$. + + Damit folgt mit der 2. Cramerschen Regel: + \[ + A^{-1} = \frac{1}{|A|} \tilde{A} \in G + .\] + \end{enumerate} + \end{proof} + \item Beh.: $G$ ist nicht abelsch. + \begin{proof} + \begin{align*} + \begin{pmatrix} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 0 & 1 & 1 + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 1 & 1 + \end{pmatrix} + = + \begin{pmatrix} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 0 & 2 & 1 + \end{pmatrix} + \neq + \begin{pmatrix} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 1 & 2 & 1 + \end{pmatrix} + = + \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 1 & 1 + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 0 & 1 & 1 + \end{pmatrix} + .\end{align*} + \end{proof} + \item Beh.: $LU$ Zerlegung eindeutig. + \begin{proof} + Sei $A \in \mathbb{K}^{n \times n}$ regulär und + $A = LU = \tilde{L}\tilde{U}$ mit $L, \tilde{L} \in G$ und + $U, \tilde{U}$ obere Dreicksmatrizen. + + Dann ist zunächst $U, \tilde{U}$ regulär, denn: $L, \tilde{L} \in G$, also regulär und + die Menge der regulären Matrizen in $\mathbb{K}^{n \times n}$ Gruppe. Somit + \[ + A = LU = \tilde{L} \tilde{U} \implies L^{-1}A = U \land \tilde{L}^{-1}A = \tilde{U} + .\] + Damit folgt + \begin{align*} + &A = LU \\ + \implies & A = \tilde{L}\tilde{U} (\tilde{L}\tilde{U})^{-1} LU \\ + \implies & A = A (\tilde{L}\tilde{U})^{-1} LU \\ + \implies & E_n = (\tilde{L}\tilde{U})^{-1} LU \\ + \implies & E_n = \tilde{U}^{-1} \tilde{L}^{-1} L U \\ + \implies &\tilde{U}U^{-1} = \tilde{L}^{-1}L \in G + .\end{align*} + Da $\tilde{U}$ und $U^{-1}$ obere Dreiecksmatrizen, ist auch das Produkt, + analog zu (a) eine obere Dreicksmatrix, d.h. + $\tilde{U}U^{-1} \in G$ ist obere Dreicksmatrix, damit folgt + $\tilde{U}U^{-1} = E_n$, also $\tilde{L}^{-1}L = E_n \implies L = \tilde{L}$. Also + $LU = A = L \tilde{U}$. Da $L$ regulär, folgt $U = \tilde{U}$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}