From 52281b15256442bfc7b1b8083034dd7e6b028adc Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Thu, 14 May 2020 21:23:16 +0200 Subject: [PATCH] ana7 done, fixed all bad breaks and overfull boxes --- ana1.pdf | Bin 339651 -> 317567 bytes ana1.tex | 22 +++--- ana4.pdf | Bin 224586 -> 265330 bytes ana4.tex | 13 ++-- ana5.pdf | Bin 163157 -> 191984 bytes ana5.tex | 10 +-- ana6.pdf | Bin 218557 -> 273418 bytes ana6.tex | 6 +- ana7.pdf | Bin 0 -> 284912 bytes ana7.tex | 179 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 531394 -> 592508 bytes analysisII.tex | 1 + 12 files changed, 206 insertions(+), 25 deletions(-) create mode 100644 ana7.pdf create mode 100644 ana7.tex diff --git a/ana1.pdf b/ana1.pdf index 5c520c57afe95bcccb592891f3a5d28bccd4e240..f9831dfa475829faef8f01d54b299740d5a7a095 100644 GIT binary patch delta 89032 zcmZ5mV|3n6ux^_)Mq@i^Y};zwH)w3z&abiUH)w1$d1E)W+1R#y`@i?w{WLp!X3y-I z^X$ydv)i+Z`TiRNjY?TUijkFx6M<@Swr>%El_eex1B#oCGts}77SNV;UTHw@IM$eK 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\forall k,l \geq N_{\varepsilon}:\norm{x^{(k)} - x^{(l)}}_{\infty} < \varepsilon$. Betrachte Komponentenfolge $(x_{i}^{(k)})_{k \in \N}, i = 1,...,n$. Die Komponentenfolgen sind Cauchy-Folgen, weil $$\left| x_{i}^{(k)} - x_{i}^{(l)} \right| \leq \norm{x^{(k)} - x^{(l)}}_{\infty} < \varepsilon, \forall k,l \geq N_{\varepsilon}, \forall i = 1,...,n.$$ $\implies \lim_{k \to \infty} x_{i}^{(k)} \eqqcolon x_{i} \implies x^{(k)} \overset{k \to \infty}{\longrightarrow} x = \icol{x_{1} \\ \vdots \\ x_{n}}$ in $\ell_{\infty}$ Norm. \item Sei $(x^{(k)})_{k \in \N}$ beschränkt $\implies$ $(x_{i}^{(k)})_{k \in \N}, \forall i = 1,...,n$ auch beschränkt \\ @@ -202,7 +201,7 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{enumerate} \end{bsp} -\begin{satz}[Eigenschaften abgeschlossener Mengen] +\begin{satz}[Eigenschaften abgeschlossener Mengen]\ \begin{enumerate}[(1)] \item Sind $V, U$ ($V,U \subset \K^{n}$) abgeschlossen, dann ist $U \cup V \subset \K^{n}$ auch abgeschlossen. \item Sind $U_{i}, (i \in I)$ abgeschlossene Mengen in $\K^{n}$. 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\end{satz} @@ -80,12 +80,12 @@ \begin{definition}[Inneres, Abschluss] Sei $M \subset \mathbb{K}^{n}$ - \begin{itemize} + \begin{itemize}\vspace*{-3mm} \item Die Menge $M^{\circ} := M \setminus \partial M$ heißt das \underline{Innere} von $M$. \item Die Menge $\overline{M} := M \cup \partial M$ heißt der \underline{Abschluss} von $M$. - \end{itemize} + \end{itemize}\vspace*{-3mm} \end{definition} \begin{satz}[Inneres ist Offen, Abschluss ist abgeschlossen] @@ -153,7 +153,7 @@ \begin{definition}[Kompaktheit] Eine Menge $M \subset \mathbb{K}^{n}$ heißt \underline{kompakt} - (bzw. \underline{folgenkompakt}), wenn jede Folge aus $M$ eine + (\underline{folgenkompakt}), wenn jede Folge aus $M$ eine konvergente Teilfolge mit Grenzwert in $M$ besitzt. \end{definition} diff --git a/ana6.pdf b/ana6.pdf index 3093e402fabaa164b3fbc2edf82d0bf34133e031..67a82d85efe2c6f9c58f2b564576df86a9bd5970 100644 GIT binary patch delta 151410 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DtaF19 diff --git a/ana6.tex b/ana6.tex index b80aa2d..8d3a81d 100644 --- a/ana6.tex +++ b/ana6.tex @@ -85,11 +85,11 @@ Wichtige Ungleichungen \begin{lemma}[Ungleichung von Hölder] Seien $p,q \in \R, p>1, \ q<\infty, \ \frac{1}{p} + \frac{1}{q} = 1$. Dann gilt $$\underbrace{|(x,y)_2|}_{% - \footnotesize\begin{tabular}{c}euklidisches\\Skalarprodukt\end{tabular}} + \text{\footnotesize\begin{tabular}{c}euklidisches\\Skalarprodukt\end{tabular}}} \leq \underbrace{\norm{x}_p}_{% - \footnotesize\begin{tabular}{c}$\ell_p$-Norm\\ von $x$\end{tabular}} + \text{\footnotesize\begin{tabular}{c}$\ell_p$-Norm\\ von $x$\end{tabular}}} \cdot \underbrace{\norm{y}_q}_{% - \footnotesize\begin{tabular}{c}$\ell_q$-Norm\\ von $y$\end{tabular}} $$ + \text{\footnotesize\begin{tabular}{c}$\ell_q$-Norm\\ von $y$\end{tabular}}}$$ \end{lemma} \begin{proof} diff --git a/ana7.pdf b/ana7.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0e2247015c201090ef8535e0dd1c96ca711f4123 GIT binary patch literal 284912 zcma(2Qnvj8UF|ccmAKPSYv|Gd&DdmR67JD6_56(@Du!;6S#1siMXGapgi)Hjgjh@qCN2 zOr>P`c;Vxtsg5Av?dyqhE*r9JQK8o-`gcBgOre+m`(OkUF%Kq*j<~~cmS$a$50E~C zq_2sQ^T(!JEU4#UVEejklJx_6yjra1EQ6wIaJ|%&-R537U~Z 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zpbroRJB5OwwRGJeNPAiYu-%0q9bu=?Ff0rWPTDaTY;ba$TJ5$ad>P0mK96HOFRQ%eW0gCpXI#wNxDthN@x1dZ3y)FK$d{vHCZV*v#Efp!Xq)&9oE|A3H{ z9_d;;P$avT!}}Yn^0-F}xU$+e6dUqIx!kFH>WZb&a729kMTNF|X^90Q|K!eNQ-Gv4&qf6vN<5ThkxhF4%j;j~&BUwacM}S)hadtMVp^&t);jsA z+&y58he5HPe3K`d@)C6&QaX^HDd~_x7yV-~j|Ta3f`ps$E=BJ0ht`W$roh1Fz;qUU)3OLXnm%JT0vJV_LencXOqveu^Dxc& zxqH(jE9L6eMYes`nSFCVLB{^iM{yN)RlrzvFAM;snHRRR& xZzGviSPx_wYSVw3lli75WHi`H@B7=yVpEvxF!0X=g3-ZY(7F&MB@=5R 0$ (Definitheit von Normen). Sei $$\norm{A}_\infty = \max\limits_{1\leq i\leq n} \sum_{j = 1}^{n}|a_{ij}| = \sum_{j = 1}^{n}|a_{mj}|\text{ für ein }m\in \{1, \dots, n\}.$$ Setze $z_j = \frac{|a_{mj}|}{a_{mj}}$, falls $a_{mj} \neq 0$ und sonst $z_j = 0$. ($z_j = \operatorname{sign}(a_{mj})$). Für $z = \begin{pmatrix} + z_1 \\ \vdots \\ z_n + \end{pmatrix}$ gilt dann $\norm{z}_\infty = 1$ und $$(Az)_m = \sum_{j = 1}^{n}a_{mj}z_j = \sum_{j = 1}^{n}|a_{mj}| = \norm{A}_\infty.$$ Es folgt + $$\norm{A}_\infty = (Az)_m \leq \norm{Az}_\infty \leq \sup\limits_{\norm{y}_\infty = 1} \norm{Ay}_\infty \leq \sup\limits_{\norm{y}_\infty = 1} \norm{A}_\infty \cdot \underbrace{\norm{y}_\infty}_{=1} = \norm{A}_\infty$$ + $$\implies \norm{A}_\infty = \sup\limits_{\norm{y}_\infty = 1} \norm{Ay}_\infty$$ + \end{enumerate} + Beweis für $l_1$ analog. +\end{proof} +\begin{definition} + \begin{enumerate} + \item Eigenwerte $\lambda \in \K$ einer Matrix $A\in \K^{n\times n} = $ Nullstellen des charakteristischen Polynoms + $$p(\lambda) = \det(A - \lambda \mathbb{I})$$ + \item $\sigma(A) = \{\lambda |\lambda\text{ Eigenwert von } A\}$ heißt \textbf{Spektrum} von $A$. + \item $\forall \lambda \in \sigma(A) \exists \text{Eigenvektor } w\in \K^n\setminus\{0\}:$ + $$A w = \lambda w$$ + Die Eigenvektoren zu $\lambda$ bilden einen Vektorraum, den \textbf{Eigenraum} zu $\lambda$ mit Dimension = \textbf{geometrische Vielfachheit} von $\lambda$. + \item Abschätzung der Eigenwerte: Sei $\lambda \in \sigma(A)$ und $w$ ein Eigenvektor zu $\lambda$ mit $\norm{w} = 1$. + Dann $|\lambda| = |lambda| \cdot \norm{w} = \norm{\lambda w} = \norm{Aw} \underset{\text{Verträglichkeit}}{\leq} \norm{A}\cdot \norm{w} = \norm{A} \implies |\lambda| \leq \norm{A}$ + \item $A$ heißt \textbf{hermitesch}, falls gilt + $$A = \overline{A}^T\quad (a_{ij} = \overline{a_{ji}})$$ + Reelle hermitesche Matrizen heißen \textbf{symmetrisch}. Für das Skalarproukt gilt + $$A = \overline{A}^T \Leftrightarrow (Ax, y)_2 = (x, Ay)_2 \forall x, y \in \K^n$$ + Hermitesche Matrizen sind diagonalisierbar = ähnlich zu einer Diagonalmatrix, alle Eigenwerte einer hermiteschen Matrix sind reell. $\exists$ eine orthonormale Basis aus Eigenvektoren. + \item $A\in \K^{n\times n}$ heißt \textbf{positiv definit}, wenn gilt $(Ax, x)_2 \in \R$, $(Ax, x)_2 > 0\forall x \in \K^n\setminus\{0\}$. Eine hermitesche Matrix ist positiv definit $\Leftrightarrow$ alle Eigenwerte sind positiv. + \item $\norm{\cdot}_2$ ($l_2$-Norm) im $\K^n$ erzeugt eine natürliche \textbf{Matrixnorm(Spektralnorm)} $\norm{\cdot}_2$ + \end{enumerate} +\end{definition} +\begin{lemma}[Spektralnorm] + Sei $A \in \K^{n\times n}$. Dann ist $\overline{A}^TA \in \K^{n\times n}$ hermitesch und positiv semidefinit. Für die Spektralnorm gilt + $$\norm{A}_2 = \max \{\sqrt{|\lambda|}, \lambda \in \sigma(\overline{A}^TA)\}$$ + Sei $A$ hermitesch, bzw. symmetrisch, dann gilt $\norm{A}_2 = \max\{\sqrt{|\lambda|}, \lambda\in \sigma(A)\}$ +\end{lemma} +\begin{proof} + später +\end{proof} +\begin{definition}[orthonormale/unitäre Matrizen] + Eine Matrix $Q\in \K^{m\times n}$ heißt \textbf{orthonormal}, wenn ihre Spaltenvektoren ein Orthonormalsystem im $\K^m$ bilden, d.h. + $$Q = (q_1, \dots, q_n)\qquad q_j \in \K^m$$ + $$(q_i, q_j)_2 = \sum_{k = 1}^{m} q_{ik}\cdot \overline{q_{kj}} = \begin{cases} + 1, &i =j\\ + 0, &\text{sonst} + \end{cases}$$ + Falls $m =n$ heißt $Q$ unitär. +\end{definition} +\begin{lemma} + Sei $Q\in \K^{n\times n}$ unitär. Dann ist $Q$ regulär, $Q^{-1} = \overline{Q}^T$ und $(Qx, Qy)_2 = (x,y)_2,\; x,y\in \K^n$ + $$\norm{Qx}_2 = \norm{x}_2, \; x\in \K^n$$ + d.h. euklidisches Skalarproukt und euklidische Norm sind invariant unter unitären Transformationen und folglich $\norm{Q}_2 = \norm{Q^{-1}}_2 = 1$ +\end{lemma} +\begin{proof} + \begin{enumerate} + \item Z.Z. $Q^{-1} = \overline{Q}^T$\\ + Sei $Q = (q_1, \dots, q_n), \overline{Q}^T = \begin{pmatrix} + \overline{q}_1^T\\ + \dots\\ + \overline{q}_n^T + \end{pmatrix}$. Dann gilt + $$\overline{Q}^T \cdot Q = \begin{pmatrix} + \overline{q}_1^T \cdot q_1 & \dots & \overline{q}_1^T \cdot q_n\\ + \vdots & & \vdots\\ + \overline{q}_n^T \cdot q_1 & \dots & \overline{q}_n^T \cdot q_n + \end{pmatrix} \overset{Q \text{ unitär}}{=} \begin{pmatrix} + 1 & & 0\\ + & \ddots & \\ + 0 & & 1 + \end{pmatrix}$$ + \item \begin{align*} + (Qx, Qy)_2 &= (x, \overline{Q}^TQ y)_2 = (x,y)_2\\ + \norm{Qx}_2^2 &= (Qx, Qx)_2 = (x,x)_2 = \norm{x}_2^2\\ + \norm{Q}_2 &= \sup\limits_{\norm{x_2} =1} \norm{Qx}_2 = \sup\limits_{\norm{x_2} =1} \norm{x}_2 = 1\\ + \norm{Q^{-1}}_2 &= \sup\limits_{\norm{x_2} =1} \norm{Q^{-1}x}_2 = \sup\limits_{\norm{x_2} =1} \norm{QQ^{-1}x}_2 = \sup\limits_{\norm{x_2} =1} \norm{x}_2 = 1 + \end{align*} + \end{enumerate} +\end{proof} +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index dc4cd3187bff01b8d5083f132cb610fd01e9b566..fe54b8869b7eee5e741e22ccfcc27bd1e8772b3d 100644 GIT binary patch delta 526467 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