From 1d358546b1b5202dbd9425094fee8c5c44b13e83 Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 22 Jun 2023 22:35:59 +0200 Subject: [PATCH] feat(geometria): aggiunge indice di Witt nel caso K=C --- .../1. Introduzione al prodotto scalare.tex | 31 +++++++++++++++--- .../main.pdf | Bin 684450 -> 688151 bytes 2 files changed, 26 insertions(+), 5 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 8dff2b3..a30c38f 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 @@ -803,7 +803,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \begin{proposition} \label{prop:disuguaglianza_sottospazio_isotropo} Se $W$ è un sottospazio isotropo di $V$, allora - $\dim W \leq \frac{\dim V + \dim V^\perp}{2}$. + $\dim W \leq \floor{\frac{\dim V + \dim V^\perp}{2}}$. \end{proposition} \begin{proof} @@ -825,7 +825,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni Sostituendo allora l'equazione \eqref{eq:disuguaglianza_sottospazio_isotropo_2} nella disuguaglianza \eqref{eq:disuguaglianza_sottospazio_isotropo_1}, si ottiene che $\dim W \leq \frac{\dim V + \dim (W \cap V^\perp)}{2}$. 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. + $\dim (W \cap V^\perp) \leq \dim V^\perp$, e quindi $\dim W \leq \frac{\dim V + \dim V^\perp}{2}$. Poiché $\dim W$ è un numero naturale, vale come conseguenza la tesi. \end{proof} \begin{definition}[indice di Witt] @@ -835,12 +835,33 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \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 - $\dim W \leq \frac{1}{2} \dim V$, \\ + $\dim W \leq \floor{\frac{1}{2} \dim V}$, \\ \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 Ancora per la \textit{Proposizione \ref{prop:disuguaglianza_sottospazio_isotropo}}, vale che $W(\varphi) \leq \frac{\dim V + \dim V^\perp}{2}$. + \li Ancora per la \textit{Proposizione \ref{prop:disuguaglianza_sottospazio_isotropo}}, vale che $W(\varphi) \leq \floor{\frac{\dim V + \dim V^\perp}{2}}$. \end{remark} +\begin{proposition} + Sia $\KK = \CC$. Allora + $W(\varphi) = \floor{\frac{\dim V + \dim V^\perp}{2}}$. +\end{proposition} + +\begin{proof} + Sia $\basis$ una base di Sylvester per $V$. In particolare, detto $k := \dim V^\perp$; sia $\basis = \{ \vv 1, \ldots, \vv{n-k}, \uu 1, \ldots, \uu k \}$ ordinata in modo tale che $\vv i$ non + sia isotropo per $1 \leq i \leq n-k$ e che $\uu i$ sia invece isotropo per + $1 \leq i \leq k$. Si costruisca allora la base $\basis' = \{\vv 1 ' := \vv 1 + i \vv 2, \, \vv 2 ' := \vv 3 + i \vv 4, \ldots, \uu 1, \ldots, \uu k\}$ ottenuta prendendo in ordine quante più coppie distinte possibili di + vettori $\vec i$ e aggiungendo al vettore con indice minore della coppia il vettore + con indice maggiore moltiplicato per $i$. In questo modo si è costruita un insieme linearmente indipendente + contenente $\floor{\frac{n-k}{2}} + k = \floor{\frac{\dim V - \dim V^\perp}{2}} + \dim V^\perp = \floor{\frac{\dim V + \dim V^\perp}{2}}$. \\ + + Sia allora $W = \Span(\basis')$. I vettori della forma $\uu i$ con $1 \leq i \leq k$ sono + chiaramente già ortogonali con gli altri vettori della base. Si consideri allora + il prodotto $\varphi(\vv i', \vv j')$. Se $i \neq j$, il prodotto ha argomenti tra di + loro già ortogonali per costruzione di $\basis$; se invece $i = j$, detto $\vv i' = \vv s + i \vv {s+1}$ con $s \in \NN$, $\varphi(\vv i', \vv i') = \varphi(\vv s, \vv s) - \varphi(\vv {s+1}, \vv {s+1}) = 1 - 1 = 0$. Allora $M_{\basis'}(\restr{\varphi}{W}) = 0 \implies \restr{\varphi}{W} = 0$. Pertanto $W$ è un sottospazio isotropo di dimensione + $\floor{\frac{\dim V + \dim V^\perp}{2}}$. Poiché per la \textit{Proposizione \ref{prop:disuguaglianza_sottospazio_isotropo}} tale dimensione maggiora tutte le + dimensioni dei sottospazi isotropi, si conclude che $W(\varphi) = \floor{\frac{\dim V + \dim V^\perp}{2}}$, da cui la tesi. +\end{proof} + \begin{proposition} Sia $\KK = \RR$. Allora $W(\varphi) = \iota_0 + \min\{\iota_+(\varphi), \iota_-(\varphi)\}$. @@ -854,7 +875,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni 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 $\varphi(\vv i, \vv i) = 1$ + 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 $1 \leq i \leq b$ 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 $W = \Span(\basis')$. \\ 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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