From 00e94943128ce644dae6147e2a4a4ed2d936b1ad Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 22 Jun 2023 21:53:19 +0200 Subject: [PATCH] feat(geometria): generalizza l'indice di Witt in R al caso degenere --- .../1. Introduzione al prodotto scalare.tex | 45 ++++++++++-------- .../main.pdf | Bin 682899 -> 684450 bytes .../main.tex | 2 +- 3 files changed, 25 insertions(+), 22 deletions(-) diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex index e33cf85..8dff2b3 100644 --- a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex +++ b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex @@ -725,7 +725,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni $p + q = \rg(\restr{\varphi}{W}) = k$. \\ Siano ora $\basis_\perp$ e $\basis_\perp'$ due basi di Sylvester di $W$ e $W'$. Siano $A = M_{\basis_\perp}(\restr{\varphi}{W})$ - e $A' = M_{\basis_\perp'}(\restr{\varphi}{W})$. Allora $\det(A) = (-1)^p (-1)^q$, mentre $\det(A') = (-1)^p (-1)^q \, d'$, + e $A' = M_{\basis_\perp'}(\restr{\varphi}{W})$. Allora $\det(A) = 1^p (-1)^q$, mentre $\det(A') = 1^p (-1)^q \, d'$, dove $d' \in \{-1, 0, 1\}$. Allora $\det(A') = \det(A) d' \implies d' = \frac{\det(A')}{\det(A)}$, dal momento che $\det(A) \neq 0$, essendo $\restr{\varphi}{W}$ non degenere. \\ @@ -802,8 +802,8 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \end{remark} \begin{proposition} \label{prop:disuguaglianza_sottospazio_isotropo} - Sia $\varphi$ non degenere. Se $W$ è un sottospazio isotropo di $V$, allora - $\dim W \leq \frac{1}{2} \dim V$. + Se $W$ è un sottospazio isotropo di $V$, allora + $\dim W \leq \frac{\dim V + \dim V^\perp}{2}$. \end{proposition} \begin{proof} @@ -815,17 +815,17 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \vskip 0.05in - Inoltre, dal momento che $\varphi$ è non degenere, vale anche che: + Inoltre, per la formula delle dimensioni del prodotto scalare, vale anche che: \begin{equation} \label{eq:disuguaglianza_sottospazio_isotropo_2} - \dim W + \dim W^\perp = \dim V \implies \dim W^\perp = \dim V - \dim W. + \dim W^\perp = \dim V + \dim (W \cap V^\perp) - \dim W. \end{equation} \vskip 0.05in Sostituendo allora l'equazione \eqref{eq:disuguaglianza_sottospazio_isotropo_2} nella disuguaglianza - \eqref{eq:disuguaglianza_sottospazio_isotropo_1}, si ottiene che $\dim W \leq \frac{1}{2} \dim V$, - ossia la tesi. + \eqref{eq:disuguaglianza_sottospazio_isotropo_1}, si ottiene che $\dim W \leq \frac{\dim V + \dim (W \cap V^\perp)}{2}$. Dal momento che $W \cap V^\perp \subseteq V^\perp$, + $\dim (W \cap V^\perp) \leq \dim V^\perp$, e quindi $\dim W \leq \frac{\dim V + \dim V^\perp}{2}$, da cui la tesi. \end{proof} \begin{definition}[indice di Witt] @@ -834,33 +834,36 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \end{definition} \begin{remark}\nl + \li Se $W$ è isotropo e $\varphi$ è non degenere, il risultato della \textit{Proposizione \ref{prop:disuguaglianza_sottospazio_isotropo}} si riduce alla disuguaglianza + $\dim W \leq \frac{1}{2} \dim V$, \\ \li Se $\varphi > 0$ o $\varphi < 0$, $W(\varphi) = 0$. Infatti ogni sottospazio non nullo $W$ di $V$ non ammette vettori isotropi, da cui si deduce che $\restr{\varphi}{W} \neq 0$. \\ - \li Se $\varphi$ è non degenere, per la \textit{Proposizione \ref{prop:disuguaglianza_sottospazio_isotropo}}, vale che $W(\varphi) \leq \frac{1}{2} \dim V$. + \li Ancora per la \textit{Proposizione \ref{prop:disuguaglianza_sottospazio_isotropo}}, vale che $W(\varphi) \leq \frac{\dim V + \dim V^\perp}{2}$. \end{remark} -%TODO: dimostrare che in generale $W(\varphi) = \min\{\iota_+(\varphi), \iota_-(\varphi)\} + \iota_0$ \begin{proposition} - Sia $\KK = \RR$ e sia $\varphi$ non degenere. Allora - $W(\varphi) = \min\{\iota_+(\varphi), \iota_-(\varphi)\}$. + Sia $\KK = \RR$. Allora + $W(\varphi) = \iota_0 + \min\{\iota_+(\varphi), \iota_-(\varphi)\}$. \end{proposition} \begin{proof} - Senza perdità di generalità si assuma $\iota_-(\varphi) \leq \iota_+(\varphi)$ (il caso $\iota_-(\varphi) > \iota_+(\varphi)$ è analogo). Sia $W$ un sottospazio con $\dim W > \iota_-(\varphi)$. Sia $W^+$ - un sottospazio con $\dim W^+ = \iota_+(\varphi)$ e $\restr{\varphi}{W^+} > 0$. Allora, per la formula - di Grassmann, $n - \dim(W \cap W^+) < \dim (W + W^\perp) \leq n \implies \dim (W \cap W^+) > 0$. Quindi $\exists \w \in W$, $\w \neq \vec 0$ tale che $\varphi(\w, \w) > 0$, da cui - si ricava che $W$ non è isotropo. Pertanto $W(\varphi) \leq \iota_-(\varphi)$. \\ - - Sia $a := \iota_+(\varphi)$ e sia $b := \iota_-(\varphi)$. - Sia ora $\basis = \{ \vv 1, \ldots, \vv a, \ww 1, \ldots, \ww b \}$ una base di Sylvester per $\varphi$. Siano $\vv 1$, ..., $\vv a$ tali che $\varphi(\vv i, \vv i) = 1$ + Senza perdità di generalità si assuma $\iota_-(\varphi) \leq \iota_+(\varphi)$ (il caso $\iota_-(\varphi) > \iota_+(\varphi)$ è analogo). Sia $W$ un sottospazio con $\dim W > \iota_0(\varphi) + \iota_-(\varphi)$. Sia $W^+$ + un sottospazio con $\dim W^+ = \iota_+(\varphi)$ e $\restr{\varphi}{W^+} > 0$. Si + osserva pertanto che $\dim W + \dim W^+ > \iota_+(\varphi) + \iota_-(\varphi) + \iota_0(\varphi) = n$: allora, per la formula + di Grassmann, $n - \dim(W \cap W^+) < \dim (W + W^+) \leq n \implies \dim (W \cap W^+) > 0$. Quindi $\exists \w \in W$, $\w \neq \vec 0$ tale che $\varphi(\w, \w) > 0$, da cui + si ricava che $W$ non è isotropo. Pertanto $W(\varphi) \leq \iota_0(\varphi) + \iota_-(\varphi)$. \\ + + Siano $a := \iota_+(\varphi)$, $b := \iota_-(\varphi)$ e $c := \iota_0(\varphi)$. + Sia $\basis = \{ \vv 1, \ldots, \vv a, \ww 1, \ldots, \ww b, \UU 1, \ldots, \UU c \}$ una base di Sylvester per $\varphi$. Siano $\vv 1$, ..., $\vv a$ tali che $\varphi(\vv i, \vv i) = 1$ con $1 \leq i \leq a$. Analogamente siano $\ww 1$, ..., $\ww b$ tali che $\varphi(\ww i, \ww i) = -1$ con - $1 \leq i \leq b$. Detta allora $\basis' = \{ \vv 1 ' := \vv 1 + \ww 1, \ldots, \vv b ' := \vv b + \ww b \}$, sia $W = \Span(\basis')$. \\ + $1 \leq i \leq b$ e siano $\uu 1$, ..., $\uu c$ tali che $\varphi(\uu i, \uu i) = 0$ con + $1 \leq i \leq c$. Posto $\basis' = \{ \vv 1 ' := \vv 1 + \ww 1, \ldots, \vv b ' := \vv b + \ww b, \uu 1, \ldots, \uu c \}$, si definisca $W = \Span(\basis')$. \\ - Si osserva che $\basis'$ è linearmente indipendente, e dunque che $\dim W = \iota_-$. Inoltre + Si osserva che $\basis'$ è linearmente indipendente, e dunque che $\dim W = \iota_-(\varphi) + \iota_0(\varphi)$. Chiaramente i vettori $\uu i$ sono ancora ortogonali con gli elementi di $\basis'$ e sono tali per cui $\varphi(\uu i, \uu i) = 0$ $\forall 1 \leq i \leq c$. Inoltre $\varphi(\vv i ', \vv j ') = \varphi(\vv i + \ww i, \vv j + \ww j)$. Se $i \neq j$, allora $\varphi(\vv i ', \vv j ') = 0$, dal momento che i vettori di $\basis$ sono a due a due ortogonali tra loro. Se invece $i = j$, allora $\varphi(\vv i ', \vv j ') = \varphi(\vv i, \vv i) + \varphi(\ww i, \ww i) = 1-1=0$. Quindi $M_{\basis'}(\restr{\varphi}{W}) = 0$, da cui si conclude che $\restr{\varphi}{W} = 0$. - Pertanto $W(\varphi) \geq i_-(\varphi)$, e quindi si conclude che $W(\varphi) = i_-(\varphi)$, da cui la tesi. + Pertanto $W(\varphi) \geq \iota_0(\varphi) + \iota_-(\varphi)$, e quindi si conclude che $W(\varphi) = \iota_0(\varphi) + \iota_-(\varphi)$, da cui la tesi. \end{proof} \section{Isometrie tra spazi vettoriali} diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/4. 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Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.tex index d6d79dc..0e2e765 100644 --- a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.tex +++ b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.tex @@ -13,7 +13,7 @@ \begin{document} \title{I prodotti di uno spazio vettoriale} - \subtitle{Dispense del corso di Geometria 1} + \subtitle{Dispense del corso di Geometria 1\\(ancora in corso di correzione e revisione)} \author{Gabriel Antonio Videtta} \date{A.A. 2022/2023 \\ \vskip 1in \includegraphics[scale=0.3]{logo.png}} \maketitle