From afca5485e169ff9dfc0d60028c15d591b9c4761f Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 13 May 2023 23:34:18 +0200 Subject: [PATCH] feat(geometria): termina di aggiungere la teoria generale sulle quadriche --- .../main.pdf | Bin 203079 -> 204327 bytes .../main.tex | 2 +- .../main.pdf | Bin 174831 -> 0 bytes .../main.tex | 99 ---------- .../main.pdf | Bin 0 -> 187108 bytes .../main.tex | 187 ++++++++++++++++++ 6 files changed, 188 insertions(+), 100 deletions(-) delete mode 100644 Geometria 1/Spazi affini e coniche/2023-05-10, Classificazione delle coniche/main.pdf delete mode 100644 Geometria 1/Spazi affini e coniche/2023-05-10, Classificazione delle coniche/main.tex create mode 100644 Geometria 1/Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.pdf create mode 100644 Geometria 1/Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.tex diff --git a/Geometria 1/Spazi affini e coniche/2023-04-28, 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affini/main.tex index eecd8a4..08a4a61 100644 --- a/Geometria 1/Spazi affini e coniche/2023-04-28, Indipendenza e applicazioni affini/main.tex +++ b/Geometria 1/Spazi affini e coniche/2023-04-28, Indipendenza e applicazioni affini/main.tex @@ -343,7 +343,7 @@ \begin{definition} [applicazione lineare associata ad un'applicazione affine] Data un'applicazione affine $\varphi : E \to E'$ e dato $O \in E$, si definisce $g : V \to V'$ tale che - $g(\v) = \varphi(O + \v) - \varphi(O)$ come l'\textbf{applicazione lineare associata a $\varphi$}. + $g(\v) = \varphi(O + \v) - \varphi(O)$ come l'\textbf{applicazione lineare associata a $\varphi$} (detta anche \textit{differenziale} di $f$, ed indicata con $df$). \end{definition} \begin{remark}\nl diff --git a/Geometria 1/Spazi affini e coniche/2023-05-10, Classificazione delle coniche/main.pdf b/Geometria 1/Spazi affini e coniche/2023-05-10, Classificazione delle coniche/main.pdf deleted file mode 100644 index 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-\date{10 maggio 2023} - -\begin{document} - - \maketitle - - \begin{center} - \Large \textbf{Classificazione delle coniche} - \end{center} - - \wip - - \begin{note} - Si assume che, nel corso del documento, valga che $\Char \KK \neq 2$. - \end{note} - - \begin{definition} [quadriche] Si dice \textbf{quadrica} un qualsiasi luogo di zeri - di un polinomio $p \in \KK[x_1, \ldots, x_n]$ con $\deg p = 2$. - \end{definition} - - \begin{definition} [coniche] Si dice \textbf{conica} una quadrica relativa ad un polinomio - in due variabili. - \end{definition} - - \begin{remark}\nl - \li Una quadrica è invariante per la relazione $\sim$ su $\KK[x_1, \ldots, x_n]$, dove - $p_1 \sim p_2 \defiff \exists \alpha \in \KK^* \mid p_1 = \alpha p_2$. Infatti - il luogo di zeri di un polinomio non varia se esso viene moltiplicato per una costante non nulla di $\KK$. \\ - \li Una quadrica può essere vuota (come nel caso della conica relativa a $x^2 + y^2 + 1$ in $\RR$). \\ - \li Si identifica con la notazione $p(\x)$ con $\x \in \KK^n$, la valutazione del polinomio $p$ nelle coordinate - di $\x$. Per esempio, se $\x = (1, 2)$ e $p(x, y) = x^2 + y^2$, con $p(\x)$ si identifica il valore - $p(1, 2) = 1^2 + 2^2 = 5$. - \end{remark} - - \begin{remark} [riscrittura di $p$ mediante matrici] - Sia $p \in \KK[x_1, \ldots, x_n]$ di grado due. Allora $p$ si può sempre scrivere come $p_2 + p_1 + p_0$, - dove $p_i$ è un polinomio omogeneo contenente soltanto monomi di grado $i$. \\ - - In particolare, $p_2(x_1, \ldots, x_n)$ può essere sempre riscritto come $\sum_{i=1}^n \sum_{j=1}^n a_{ij}$ - con $a_{ij} \in \KK$ con $a_{ij} = a_{ji}$. - È infatti sufficiente "sdoppiare" il coefficiente $c_{ij}$ - di $x_i x_j$ in due metà, in modo tale che $c_{ij} x_i x_j = \frac{c_{ij}}{2} x_i x_j + \frac{c_{ij}}{2} x_i x_j = \frac{c_{ij}}{2} x_i x_j + \frac{c_{ij}}{2} x_j x_i$. Inoltre, anche $p_1(x_1, \ldots, x_n)$ può essere riscritto come $\sum_{i=1}^n b_{ij}$. \\ - - Si possono allora considerare la matrice $A \in M(n, \KK)$ ed il vettore $\vec b \in \KK^n$, definiti in modo tale che: - - \[ A = (a_{ij})_{i,j=1\mbox{--}n}, \qquad \vec b = (b_i)_{i=1\mbox{--}n} \in \KK^n. \] - - \vskip 0.05in - - Infatti, $A$ e $\vec b$ soddisfano la seguente identità: - - \[ p(\x) = \x^\top A \x + \vec b^\top \x + c, \] - - \vskip 0.05in - - che, riscritta tramite l'identificazione di $\AnK$ come l'iperpiano $H_{n+1} \in \Aa_{n+1}(\KK)$, - diventa: - - \[ p(\x) = {\hat \x}^\top \hat A \hat \x, \quad \dove \hat A = \Matrix{A & \rvline & \nicefrac{\vec b}{2} \, \\[1pt] \hline & \rvline & \\[-9pt] \nicefrac{{\vec b}^\top}{2} & \rvline & c }. \] - - \vskip 0.05in - - Si osserva che $\hat A$ è una matrice simmetrica di taglia $n+1$ a elementi in $\KK$, e in quanto - tale essa induce un prodotto scalare su $\KK^{n+1}$. Pertanto la quadrica relativa $p$ è esattamente - l'intersezione tra $H_{n+1}$ e $\CI(\hat A)$, identificando $\KK^{n+1}$ come $H_{n+1}$, ossia - la quadrica è esattamente $\iota\inv(H_{n+1} \cap \CI(\hat A))$. - \end{remark} - - \begin{definition}[matrice associata ad una quadrica] - Si definisce la costruzione appena fatta di $\hat A$ come la \textbf{matrice associata alla quadrica relativa a $p$}, e si indica con $\MM(p)$. In particolare, $A$ è detta la matrice che rappresenta la \textit{parte quadratica}, e si indica con $\AA(p)$, mentre $\nicefrac{\vec b}2$ rappresenta la \textit{parte lineare}, indicata con $\Ll(p)$, - e $c = c(p)$ è detto \textit{termine noto}. - \end{definition} - - \begin{definition}[azione di $A(\Aa_n(\KK))$ su $\KKxn$] - Sia $f \in A(\Aa_n(\KK))$. Allora $A(\Aa_n(\KK))$ agisce su $\KKxn$ in modo tale che - $p' = p \circ f$ è un polinomio per cui $p'(\x) = p(f(\x))$. - \end{definition} - - \begin{proposition} [formula del cambiamento della matrice associata su azione di $A(\Aa_n(\KK))$] - Sia $f \in A(\Aa_n(\KK))$ e sia $p \in \KK[x_1, \ldots, x_n]$ di grado due. Allora vale - la seguente identità: - - \begin{multline*} - \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M = \Matrix{M^\top \AA(p) \vec t M & \rvline & M^\top(\AA(p) \vec t + \Ll(p)) \, \\[1pt] \hline & \rvline & \\[-9pt] \, \left(M^\top(\AA(p) \vec t + \Ll(p))\right)^\top & \rvline & p(\vec t)}, \\[0.1in] - \con \hat M = \Matrix{ M & \rvline & \vec t \, \\ \hline 0 & \rvline & 1 \, }, - \end{multline*} - - \vskip 0.05in - - dove $f(\x) = M \x + \vec t$ $\forall \x \in \KK^n$ con $M \in \GL(n, \KK)$ e $\vec t \in \KK^n$. - \end{proposition} - -\end{document} diff --git a/Geometria 1/Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.pdf b/Geometria 1/Spazi affini e coniche/2023-05-10, Quadriche e classificazione affine delle coniche/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dfac0abdb4fa93be82497da58840a74e1617156c GIT binary patch literal 187108 zcmbrm1z1(V);CVKgtQU|BqXIy5K;n4hopqGfV6~2BPAd$4N`(2(xG&BDbH4+0h 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+\documentclass[11pt]{article} +\usepackage{personal_commands} +\usepackage[italian]{babel} + +\title{\textbf{Note del corso di Geometria 1}} +\author{Gabriel Antonio Videtta} +\date{10 maggio 2023} + +\begin{document} + + \maketitle + + \begin{center} + \Large \textbf{Quadriche e classificazione affine delle coniche} + \end{center} + + \wip + + \begin{note} + Si assume che, nel corso del documento, valga che $\Char \KK \neq 2$. + \end{note} + + \begin{definition} [quadriche] Si dice \textbf{quadrica} un qualsiasi luogo di zeri + di un polinomio $p \in \KK[x_1, \ldots, x_n]$ con $\deg p = 2$. + \end{definition} + + \begin{definition} [coniche] Si dice \textbf{conica} una quadrica relativa ad un polinomio + in due variabili. + \end{definition} + + \begin{remark}\nl + \li Una quadrica è invariante per la relazione $\sim$ su $\KK[x_1, \ldots, x_n]$, dove + $p_1 \sim p_2 \defiff \exists \alpha \in \KK^* \mid p_1 = \alpha p_2$. Infatti + il luogo di zeri di un polinomio non varia se esso viene moltiplicato per una costante non nulla di $\KK$. \\ + \li Una quadrica può essere vuota (come nel caso della conica relativa a $x^2 + y^2 + 1$ in $\RR$). \\ + \li Si identifica con la notazione $p(\x)$ con $\x \in \KK^n$, la valutazione del polinomio $p$ nelle coordinate + di $\x$. Per esempio, se $\x = (1, 2)$ e $p(x, y) = x^2 + y^2$, con $p(\x)$ si identifica il valore + $p(1, 2) = 1^2 + 2^2 = 5$. + \end{remark} + + \begin{remark} [riscrittura di $p$ mediante matrici] + Sia $p \in \KK[x_1, \ldots, x_n]$ di grado due. Allora $p$ si può sempre scrivere come $p_2 + p_1 + p_0$, + dove $p_i$ è un polinomio omogeneo contenente soltanto monomi di grado $i$. \\ + + In particolare, $p_2(x_1, \ldots, x_n)$ può essere sempre riscritto come $\sum_{i=1}^n \sum_{j=1}^n a_{ij}$ + con $a_{ij} \in \KK$ con $a_{ij} = a_{ji}$. + È infatti sufficiente "sdoppiare" il coefficiente $c_{ij}$ + di $x_i x_j$ in due metà, in modo tale che $c_{ij} x_i x_j = \frac{c_{ij}}{2} x_i x_j + \frac{c_{ij}}{2} x_i x_j = \frac{c_{ij}}{2} x_i x_j + \frac{c_{ij}}{2} x_j x_i$. Inoltre, anche $p_1(x_1, \ldots, x_n)$ può essere riscritto come $\sum_{i=1}^n b_{ij}$. \\ + + Si possono allora considerare la matrice $A \in M(n, \KK)$ ed il vettore $\vec b \in \KK^n$, definiti in modo tale che: + + \[ A = (a_{ij})_{i,j=1\mbox{--}n}, \qquad \vec b = (b_i)_{i=1\mbox{--}n} \in \KK^n. \] + + \vskip 0.05in + + Infatti, $A$ e $\vec b$ soddisfano la seguente identità: + + \[ p(\x) = \x^\top A \x + \vec b^\top \x + c, \] + + \vskip 0.05in + + che, riscritta tramite l'identificazione di $\AnK$ come l'iperpiano $H_{n+1} \in \Aa_{n+1}(\KK)$, + diventa: + + \[ p(\x) = {\hat \x}^\top \hat A \hat \x, \quad \dove \hat A = \Matrix{A & \rvline & \nicefrac{\vec b}{2} \, \\[1pt] \hline & \rvline & \\[-9pt] \nicefrac{{\vec b}^\top}{2} & \rvline & c }. \] + + \vskip 0.05in + + Si osserva che $\hat A$ è una matrice simmetrica di taglia $n+1$ a elementi in $\KK$, e in quanto + tale essa induce un prodotto scalare su $\KK^{n+1}$. Pertanto la quadrica relativa $p$ è esattamente + l'intersezione tra $H_{n+1}$ e $\CI(\hat A)$, identificando $\KK^{n+1}$ come $H_{n+1}$, ossia + la quadrica è esattamente $\iota\inv(H_{n+1} \cap \CI(\hat A))$. + \end{remark} + + \begin{definition}[matrice associata ad una quadrica] + Si definisce la costruzione appena fatta di $\hat A$ come la \textbf{matrice associata alla quadrica relativa a $p$}, e si indica con $\MM(p)$. In particolare, $A$ è detta la matrice che rappresenta la \textit{parte quadratica}, e si indica con $\AA(p)$, mentre $\nicefrac{\vec b}2$ rappresenta la \textit{parte lineare}, indicata con $\Ll(p)$, + e $c = c(p)$ è detto \textit{termine noto}. + \end{definition} + + \begin{definition}[azione di $A(\Aa_n(\KK))$ su $\KKxn$] + Sia $f \in A(\Aa_n(\KK))$. Allora $A(\Aa_n(\KK))$ agisce su $\KKxn$ in modo tale che + $p' = p \circ f$ è un polinomio per cui $p'(\x) = p(f(\x))$. + \end{definition} + + \begin{definition}[equivalenza affine tra polinomi] + Si dice che due polinomi $p_1$, $p_2 \in \KKxn$ sono affinemente equivalenti se e solo se $\exists f \in A(\AnK) \mid p_1 = p_2 \circ f$. + In tal caso si scrive che $p_1 \sim p_2$. + \end{definition} + + \begin{remark}\nl + \li L'equivalenza affine è una relazione di equivalenza. \\ + \li Sia $Z(p)$ il luogo di zeri di $p$. Allora, $p_1 \sim p_2 \implies + \exists f \in A(\AnK) \mid Z(p_2) = f(Z(p_1))$. \\ + \li In generale, se $p_1 = p_2 \circ f$, vale che $Z(p_2) = f(Z(p_1))$. + \end{remark} + + \begin{proposition} [formula del cambiamento della matrice associata su azione di $A(\Aa_n(\KK))$] + Sia $f \in A(\Aa_n(\KK))$ e sia $p \in \KK[x_1, \ldots, x_n]$ di grado due. Allora vale + la seguente identità: + + \begin{multline*} + \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M = \Matrix{M^\top \AA(p) M & \rvline & M^\top(\AA(p) \vec t + \Ll(p)) \, \\[1pt] \hline & \rvline & \\[-9pt] \, \left(M^\top(\AA(p) \vec t + \Ll(p))\right)^\top & \rvline & p(\vec t)}, \\[0.1in] + \con \hat M = \Matrix{ M & \rvline & \vec t \, \\ \hline 0 & \rvline & 1 \, }, + \end{multline*} + + \vskip 0.05in + + dove $f(\x) = M \x + \vec t$ $\forall \x \in \KK^n$ con $M \in \GL(n, \KK)$ e $\vec t \in \KK^n$. + \end{proposition} + + \begin{proof} + Per definizione, $p \circ f$ è tale che $(p \circ f)(\x) = p(f(\x)) = + p(M\x + \vec t)$. In particolare, $(p \circ f)(\x) = \widehat{\left( M \x + \vec t \right)^\top} \MM(p) \widehat{\left( M \x + \vec t \right)} = \left( \hat M \hat x \right)^\top \!\! \MM(p) \left( \hat M \hat x \right)$. Pertanto vale che: + + \[ (p \circ f)(\x) = \hat x^\top \hat M^\top \MM(p) \hat M \hat x \implies \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M, \] + + \vskip 0.05in + + da cui la tesi. + \end{proof} + + \begin{remark}\nl + \li Per la proposizione precedente, due matrici, associate a due + polinomi di secondo grado affinemente equivalenti, variano + per congruenza, così come le matrici della parte quadratica. \\ + + Pertanto $\rg(\MM(p \circ f)) = \rg(\MM(p))$, come $\rg(\AA(p \circ f)) + = \rg(\AA(p))$ (così come, per $\KK=\RR$, non variano i segni + dei vari determinanti). Allo stesso + tempo, la classe di equivalenza di $\MM(p)$ è rappresentata completamente per $\KK = \CC$ (tramite il rango) e per $\KK = \RR$ + (tramite la segnatura), per il teorema di Sylvester. \\ + + \li Se $f$ è una traslazione, $M = I_n$, e dunque la formula + si riduce alla seguente: + + \[ \MM(p \circ f) = \Matrix{\AA(p) & \rvline & \AA(p) \vec t + \Ll(p) \, \\[1pt] \hline & \rvline & \\[-9pt] \, \left(\AA(p) \vec t + \Ll(p)\right)^\top & \rvline & p(\vec t)}. \] + + \vskip 0.05in + + In particolare, non varia la matrice relativa alla parte quadratica, + ossia vale che $\AA(p \circ f) = \AA(p)$. \\ + + \li Se $\lambda \in \KK^*$, $\MM(\lambda p) = \lambda \MM(p)$, dal + momento che $\AA(\lambda p) = \lambda \AA(p)$, così come + $\Ll(\lambda p) = \lambda \Ll(p)$ e $c(\lambda p) = \lambda c(p)$. + Tuttavia, a differenza del cambio di matrice per equivalenza + affine, per $\KK=\RR$ la segnatura non è più un invariante (infatti, in generale $\sigma(-S) = (\iota_-(S), \iota_+(S), \iota_0(S))$, se $S \in \Sym(n, \RR)$). Ciononostante non varia, in valore assoluto, la differenza tra l'indice di positività + e quello di negatività, ossia $S(\MM(p)) := \abs{\iota_+ - \iota_-}$ continua ad essere invariante. + \end{remark} + + \begin{definition} [quadrica non degenere] + Una quadrica relativa a $p \in \KKxn$ si dice \textbf{non degenere} se $\rg(\MM(p)) = n+1$ (ossia se $\det(\MM(p)) \neq 0$), e altrimenti + si dice degenere. In particolare, una conica si dice \textit{non degenere} se $\rg(\MM(p)) = 3$ e degenere altrimenti. + \end{definition} + + \begin{definition} [quadrica a centro] + Una quadrica $C$ relativa a $p \in \KKxn$ (o $p$ stesso) si dice \textbf{a centro} se + $\exists \x_0 \in \KK^n \mid p(\x_0 + \x) = p(\x_0 - \x)$ $\forall \x \in \KK^n$. In particolare, si dice che tale $\x_0$ è un \textbf{centro di simmetria} per $C$. + \end{definition} + + \begin{remark}\nl + \li Si osserva che $\vec 0$ è un centro di simmetria per $p$ se + $p(\x) = p(-\x)$, ossia se e solo se la parte lineare $\Ll(p)$ è + nulla. \\ + + \li Allora $\x_0$ è un centro di simmetria per $p$ se e solo se $\vec 0$ + è un centro di simmetria per $p \circ f$, dove $f$ è la traslazione + che manda $\vec 0$ in $\x_0$. Infatti, in tal caso, vale che $f(\x) = \x + \x_0$ e che: + + \[ (p \circ f)(\x) = p(\x + \x_0) = p(\x - \x_0) = (p \circ f)(-\x). \] + + \vskip 0.05in + + \li Per le osservazioni precedenti, vale allora che $\x_0$ è un centro + di simmetria per $p$ se e solo se la parte lineare di $p \circ f$ + è nulla, ossia se e solo se $\x_0$ è tale che $\AA(p) \x_0 + \Ll(p)$. + Pertanto $p$ è a centro se e solo se il sistema $\AA(p) \x = - \Ll(p)$ + è risolvibile, e quindi se e solo se $\rg\!\Matrix{\AA(p) & \rvline & \Ll(p)} = \rg(\AA(p))$ $\iff \Ll(p) \in \Im(\AA(p))$, per il teorema di Rouché-Capelli. Vale + dunque che $p$ è sempre a centro, se $\AA(p)$ è invertibile. \\ + + Poiché i centri di una conica sono esattamente le soluzioni del + sistema lineare $\AA(p) \x = - \Ll(p)$, essi formano un sottospazio + affine. In particolare, se $\x_0$ è un centro, vale che tale sottospazio + è esattamente $\x_0 + \Ker \AA(p)$. Pertanto, se $\AA(p)$ è invertibile + (ossia se è iniettiva), il centro è unico. + \end{remark} +\end{document}