From 80ff07289714c38905dafb3fb8a95b4b2b3ed235 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 6 May 2023 13:07:49 +0200 Subject: [PATCH] feat(geometria): aggiunge l'indice di Witt e la teoria sui sottospazi isotropi --- .../main.pdf | Bin 237510 -> 237510 bytes .../1. Introduzione al prodotto scalare.tex | 73 ++++++++++++++++++ .../main.pdf | Bin 433796 -> 442624 bytes 3 files changed, 73 insertions(+) diff --git a/Geometria 1/Prodotto scalare e hermitiano/Appunti originali/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/Appunti originali/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.pdf index 20a7c1a3e77db66d394880ec411e0b65b018c83a..f521139495ee798e3698ff47ff035ac8a2aeb9f5 100644 GIT binary patch delta 135 zcmX@MpYPayzJ?aY7N#xC@1|%P85VBlhCXy$6{R delta 135 zcmX@MpYPayzJ?aY7N#xC@1|%PS{j%dnra&usT&xmYjWxP=BKzMmZU0ZxL6q(7#SOw z0;R!9wu?_?4&ZWfaWXbCH?}l$bTTzIG&Xf~GB!20Ft#u;HFPvJbau5 0$ $\forall 1 \leq i \leq n$. Analogamente $\varphi$ è definito negativo se e solo se $\iota_- = n$, e quindi se e solo se vi sono solo variazioni di segno $\iff d_i > 0$ se $i$ è pari e $d_i < 0$ se $i$ è dispari $\iff (-1)^i \, d_i > 0$, $\forall 1 \leq i \leq n$. +\end{proof} + +\subsubsection{Sottospazi isotropi e indice di Witt} + +\begin{definition}[sottospazio isotropo] + Sia $W$ un sottospazio di $V$. Allora si dice che $W$ è un \textbf{sottospazio isotropo} di $V$ + se $\restr{\varphi}{W} = 0$. +\end{definition} + +\begin{remark}\nl + \li $V^\perp$ è un sottospazio isotropo di $V$. \\ + \li $\vec{v}$ è un vettore isotropo $\iff$ $W = \Span(\vec v)$ è un sottospazio isotropo di $V$. \\ + \li $W \subseteq V$ è isotropo $\iff$ $W \subseteq W^\perp$. +\end{remark} + +\begin{proposition} \label{prop:disuguaglianza_sottospazio_isotropo} + Sia $\varphi$ non degenere. Se $W$ è un sottospazio isotropo di $V$, allora + $\dim W \leq \frac{1}{2} \dim V$. +\end{proposition} + +\begin{proof} + Poiché $W$ è un sottospazio isotropo di $V$, vale che $W \subseteq W^\perp$. Allora vale che: + \begin{equation} \label{eq:disuguaglianza_sottospazio_isotropo_1} + \dim W \leq \dim W^\perp. + \end{equation} + + \vskip 0.05in + + Inoltre, dal momento che $\varphi$ è non degenere, 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. + \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. +\end{proof} + +\begin{definition}[indice di Witt] + Si definisce l'\textbf{indice di Witt} $W(\varphi)$ di $(V, \varphi)$ + come la massima dimensione di un sottospazio isotropo di $V$. +\end{definition} + +\begin{remark}\nl + \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 $W(\varphi) \leq \frac{1}{2} \dim V$. +\end{remark} + +\begin{proposition} + Sia $\KK = \RR$ e sia $\varphi$ non degenere. Allora + $W(\varphi) = \min\{\iota_+(\varphi), \iota_-(\varphi)\}$. +\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, $n \geq \dim (W + W^+) = \dim W + \dim W^+ - \dim (W \cap W^+) > n - \dim (W \cap W^+) \implies \dim (W \cap W^+) > 0$. 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 $\varphi(\vv i, \vv i) = 1$ + 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 $W = \Span(\basis')$. \\ + + Si osserva che $\basis'$ è linearmente indipendente, e dunque che $\dim W = \iota_-$. 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. \end{proof} \ No newline at end of file diff --git a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf index 01706176e13074b26d7acb3ef3666ea9eb2c3a9b..22128e0f010cf1d2c7ebfa37d283fde6321f5138 100644 GIT binary patch delta 94378 zcmbSy1yodB8!j_QcXun&Ff+s;DIo$1NSBmIBaKQ8-HkAW(jWp7N=Qo!NFxmj2%?m9 z!##k0Kl=UuTKC?ySi_lp_Bng+XFu<|-x~iqYHwc}?=nz0BND_9BDn&Azy*-^UZS!g zyBI;Z2<=djvVw3@%ffD^j*O8Kd}4GvKbD7Hp7|2*jUGhF!Lqs+Ow!pG z^RAh;I_fP2Juv%{`aKwK4c6Bh=D)u&>Vkm2Y=)Rbz;kWr}?W3xLFR!s~QTyR( z&zW{y9e8#{`i-`gPhytKc;x5}tlTv^}~TyxBGVb=VoDhxP^5mPE;q{ey>* z15}+`?=1!^K%Fj@H=Ht5{3WG2y2^cHRGX)R6g09F9`+qw5fnIzePf#rTQj<^KPK%) 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