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zJ#*R2_}r|Osu{TYdKH8@H4RF${N_x?A9?JCioBWOiTZA9RYfmmOQ=|h%RB6!T@|;B v4rS@6nw<|svz=_GR6_eUX8(K2U0h9_T|J$j#~Q%R3E&1`(bG#ROJV&7212|` literal 0 HcmV?d00001 diff --git a/sose2020/la/uebungen/la5.tex b/sose2020/la/uebungen/la5.tex new file mode 100644 index 0000000..a81d308 --- /dev/null +++ b/sose2020/la/uebungen/la5.tex @@ -0,0 +1,201 @@ +\documentclass[uebung]{../../../lecture} + +\title{Lineare Algebra II: Übungsblatt 5} +\author{Dominik Daniel, Christian Merten} + +\usepackage[]{gauss} + +\begin{document} + +\punkte[18] + +\begin{aufgabe} + Sei $K$ ein Körper, $n \in \N$, $A \in M_{n,n}(K)$ und + $I_A := \{ f \in K[t] \mid f(A) = 0\} $. + \begin{enumerate}[(a)] + \item Beh.: $I_A \subseteq K[t]$ ist ein Ideal und $\chi_{A}^{\text{char}} \in I_A$. + \begin{proof} + $I_A$ ist Ideal, denn + \begin{enumerate}[label=(I\arabic*)] + \item $0 \in I_A$, denn $0(A) = 0$. + \item Seien $f, g \in I_A$. Dann ist $(f+g)(A) = f(A) + g(A) = 0 + 0 = 0$, also + $f+g \in I_A$. + \item Sei $f \in I_A$ und $r \in K[t]$. Dann ist $(f \cdot r)(A) = f(A) \cdot r(A) = 0 \cdot r(a) = 0 \implies rf \in I_A$. + \end{enumerate} + Mit dem Satz von Cayley-Hamilton gilt + $\chi_{A}^{\text{char}}(A) = 0 \implies \chi_A^{\text{char}} \in I_A$. + \end{proof} + \item Beh.: Es ex. ein eindeutig bestimmtes Polynom $\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 + $I_A \neq (0)$. Wegen $K[t]$ HIR ex. ein bis auf Assoziiertheit eindeutig bestimmtes + Polynom $f \in K[t] \setminus \{0\} $ mit $(f) = I_A$. Da $K$ Körper, ex. genau + ein normiertes Polynom + $\chi_{A}^{\text{min}}$ mit $\chi_{A}^{\text{min}} \stackrel{\wedge}{=} f$ + und damit $(\chi_{A}^{\text{min}}) = I_A$. + \end{proof} + \item Sei $\lambda \in K$. Beh.: $\chi_{A}^{\text{min}}(\lambda) = 0 \iff \chi_{A}^{\text{char}}(\lambda) = 0$. + \begin{proof} + Sei $\lambda \in K$ mit $\chi_{A}^{\text{char}}(\lambda) = 0$. Dann ist $\lambda$ Eigenwert + von $A$ zu Eigenvektor $v \in K^{n} \setminus \{0\} $ mit $Av = \lambda v$. Für + $f \in K[t]$ gilt dann: + \begin{align*} + f(A)v &= a_0 v + a_1 Av + \ldots + a_m A^{m} v \\ + &= a_0 v + a_1 \lambda v + \ldots + a_m A^{m-1} \lambda v \\ + &= a_0 v + a_1 \lambda v + \ldots + a_m \lambda^{m} v \\ + &= f(\lambda) v + \intertext{Damit folgt} + 0 &= \chi_{A}^{\text{min}} (A) v = \chi_{A}^{\text{min}}(\lambda) v + .\end{align*} + Wegen $v \neq 0$, folgt $\chi_{A}^{\text{min}}(\lambda) = 0$. + + Sei nun $\chi_{A}^{\text{min}} (\lambda) = 0$. Wegen + $\chi_{A}^{\text{char}} \in (\chi_{A}^{\text{min}})$ ex. ein $f \in K[t]$ mit + $\chi_{A}^{\text{char}} = f \cdot \chi_{A}^{\text{min}}$. Damit folgt + \[ + \chi_{A}^{\text{char}}(\lambda) = f(\lambda) \cdot \chi_{A}^{\text{min}}(\lambda) = f(\lambda) \cdot 0 = 0 + .\] + \end{proof} + \item Sei $B \in M_{n,n}(K)$. Beh.: $B \approx A \implies I_B = I_A$ und + $\chi_{B}^{\text{min}} = \chi_{A}^{\text{min}} $. + \begin{proof} + Sei $B \approx A$. Dann ex. $S \in \text{GL}_n(K)$ mit $B = SAS^{-1}$. Es ist + \begin{align*} + (SAS^{-1})^{m} &= \underbrace{SA\overbrace{S^{-1} \cdot S}^{= E_n}AS^{-1} \cdot \ldots \cdot SAS^{-1}}_{m\text{-mal}} \\ + &= SA^{m}S^{-1} + \intertext{Damit folgt} + f(SAS^{-1}) &= \sum_{k=0}^{m} a_k (SAS^{-1})^{k} \\ + &= \sum_{k=0}^{m} a_k S A^{k} S^{-1} \\ + &= S \cdot \sum_{k=0}^{m} a_k A^{K} \cdot S^{-1} \\ + &= S f(A) S^{-1} + .\end{align*} + Also gilt + \begin{salign*} + & f \in I_A \\ + \iff & f(A) = 0 \\ + \stackrel{S \neq 0}{\iff} & S f(A) S^{-1} = 0 \\ + \iff& f(SAS^{-1}) = 0 \\ + \iff & f(B) = 0 + .\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, + $\chi_{A}^{\text{min}} = \chi_{B}^{\text{min}} $. + \end{proof} + \item Beh.: Mit $ A := \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ ist + $\chi_{A}^{\text{min}} \neq \chi_{A}^{\text{char}} $. + \begin{proof} + Es ist $P_A = \begin{pmatrix} t-1 & 0 \\ 0 & t-1 \end{pmatrix} $. Also + $c_1(A) = c_2(A) = t-1$, also mit 19(b) $\chi_{A}^{\text{min}} = t-1$, aber + $\chi_{A}^{\text{char}} = \text{det}(P_A) = (t-1)^2 \neq t-1 = \chi_{A}^{\text{min}} $. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $K$ ein Körper. + \begin{enumerate}[(a)] + \item Sei $g \in K[t]$ nichtkonstant und normiert mit Begleitmatrix $B_g$. Beh.: + $\chi_{B_g}^{\text{min}} = g$. + \begin{proof} + Zunächst z.Z.: $\text{deg}(\chi_{B_g}^{\text{min}} ) \ge \text{deg}(g) $. Ang. es ex. + ein $0 \neq f \in I_A$ mit $\N \ni m := \text{deg}(f) < \text{deg}(g) =: n \in \N$. Es gilt + $\forall k \in \N$ mit $k < n$: + \[ + B_g^{k}e_1 = B_g^{k-1} B_g e_1 = B_g^{k-1} e_2 = \ldots = e_{k+1} + .\] Damit folgt + \[ + f(B_g) e_1 = a_0 e_1 + a_1 e_2 + \ldots + a_m e_{m+1} = \begin{pmatrix} a_0 \\ a_1 \\ \vdots \\ a_m \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0 + .\] Also $a_0 = a_1 = \ldots = a_m = 0 \implies f = 0$ $\contr$. + Da $\chi_{A}^{\text{min}} \neq 0 \implies \text{deg}(\chi_{A}^{\text{min}} ) \ge \text{deg}(g)$. + + Da $\chi_{A}^{\text{char}} \in (\chi_{A}^{\text{min}})$, ex. ein $r \in K[t]$ + mit $g = \chi_{A}^{\text{char}} = r \cdot \chi_{A}^{\text{min}} $. Da $K[t]$ nullteilerfrei, + folgt also + $\text{deg}(g) = \text{deg}(\chi_{A}^{\text{char}} ) \ge \text{deg}(\chi_{A}^{\text{min}} ) $. + Damit folgt $\text{deg}(g) = \text{deg}(\chi_{A}^{\text{min}} ) $ und + $\chi_{A}^{\text{min}} \mid g$. Da $g$ und $\chi_{A}^{\text{min}} $ normiert, folgt + $g = \chi_{A}^{\text{min}} $. + \end{proof} + \item Seien $n \in \N$ und $A \in M_{n,n}(K)$ mit Invariantenteilern + $c_1(A), \ldots, c_n(A)$ und $c_1(A) \mid \ldots \mid c_n(A)$. Beh.: + $c_n(A) = \chi_{A}^{\text{min}} $. + \begin{proof} + Seien $g_1, \ldots, g_r$ die nichtkonstanten Invariantenteiler von $A$. Dann + ist $g_r = c_n$ und $A \approx B_{g_1, \ldots, g_r}$. $B_{g_1, \ldots, g_r}$ ist + diagonale Blockmatrix, d.h. + \[ + B_{g_1, \ldots, g_r}^{k} = \begin{pmatrix} B_{g_1}^{k} & 0 \\ + 0 & \ddots & \\ + & & B_{g_r}^{k} + \end{pmatrix} + .\] + + Damit folgt für $f \in K[t]$ beliebig. + \begin{salign*} + &\qquad f \in I_{B_{g_1, \ldots, g_r}} \\ + \iff& f(B_{g_k}) = 0 \quad \forall k = 1,\ldots,r \\ + \stackrel{\text{(a)}}{\iff}& g_k \mid f \quad \forall k = 1,\ldots,r \\ + \stackrel{g_1 \mid g_2 \mid \ldots \mid g_r}{\iff}& g_r \mid f \\ + \iff& f = q \cdot g_r \\ + \iff& f \in (g_r) + .\end{salign*} + Also folgt $(c_n) = (g_r) = I_{B_{g_1, \ldots, g_r}} \quad \stackrel{\text{18(d)}}{=} \quad I_A$. + Da $c_n$ normiert, folgt also $c_n = \chi_{A}^{\text{min}}$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es ist $n = 8$, $\chi_{A}^{\text{char}} = c_6(A) \cdot c_7(A) \cdot c_8(A) + = (t+1) \cdot t (t+1) \cdot t^2(t+1)^{3} = t^3 (t+1)^{5}$ und + $\chi_{A}^{\text{min}} = c_8(A) = t ^{3}(t+1)^{5}$. + \item Es ist $d_1(A) = \ldots = d_5(A) = 1$. $d_6(A) = c_6(A) = t+1$, + $d_7(A) = c_6(A) \cdot c_7(A) = t(t+1)^2$, $d_8(A) = d_7(A) \cdot c_8(A) = t ^3(t+1)^{5}$. + \begin{align*} + A \approx B_{c_6, c_7, c_8} = + \begin{gmatrix}[p] + -1 & 0 & 0 \\ + 0 & \begin{gmatrix}[b] 0 & 0 \\ 1 & - 1 \end{gmatrix} & 0 \\ + 0 & 0 & \begin{gmatrix}[b] 0 & 0 & 0 & 0 & 0 \\ + 1 & 0 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 & -1 \\ + 0 & 0 & 1 & 0 & -3 \\ + 0 & 0 & 0 & 1 & -3 \end{gmatrix} + \end{gmatrix} + .\end{align*} + \item Die Weistraßteiler sind + $h_1 = t+1$, $h_2 = t$, $h_3 = t+1$, $h_4 = t^2$, $h_5 = (t+1)^{3}$. + \begin{align*} + A \approx B_{h_1, h_2, h_3, h_4, h_5} = + \begin{gmatrix}[p] + -1 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & -1 & 0 & 0 \\ + 0 & 0 & 0 & \begin{gmatrix}[b] 0 & 0 \\ 1 & 0 \end{gmatrix} & 0 \\ + 0 & 0 & 0 & 0 & \begin{gmatrix}[b] 0 & 0 & -1 \\ 1 & 0 & -3 \\ 0 & 1 & -3 \end{gmatrix} + \end{gmatrix} + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Für $A = \begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix} \in M_{n,n}(\Q)$ gilt: + \begin{align*} + P_A = + \begin{pmatrix} t & -2 \\ -1 & t \end{pmatrix} \sim \begin{pmatrix} 1 & 0 \\ 0 & t^2 -2\end{pmatrix} + .\end{align*} + Also sind $c_1(A) = 1$ und $c_2(A) = t^2 -2$. Über $\Q$ ist $t^2 - 2$ irreduzibel, also + $h_1 = t^2 - 2$. Damit folgt für die Weierstraßnormalform + \begin{align*} + A \approx B_{h_1} = \begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix} + .\end{align*} + + Über $\R$ ist $t^2 - 2$ reduzibel als $t^2 - 2 = \underbrace{( t + \sqrt{2} )}_{=: \tilde{h}_1}\underbrace{( t - \sqrt{2} )}_{=: \tilde{h}_2} $, also folgt + \begin{align*} + A \approx B_{\tilde{h}_1, \tilde{h}_2} = \begin{pmatrix} \sqrt{2} & 0 \\ 0 & -\sqrt{2} \end{pmatrix} + .\end{align*} +\end{aufgabe} + +\end{document}