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zG{8WUW@8v=i)@SsCJXlbhw#C$$`-^CiMbh!z9fr&FrdQhp#)$QJB~1M1pjgH$NUGb zC?sYx3`8Nj5rLXVvP&zN2uVJhEfnN2t1Dt4kl2}np)?q|*gHW<9y_=&l;TOyiow{g zGWx3qfSDR+mum5a5qoF_trtbb8KFI(fB+C<7Yi}Qr zI7EgCxPbSczgctrW>}GYTNJj05gXDL 0$), e $W$ è + un sottospazio di $V$, vale la seguente decomposizione: + + \[ V = W \oplus^\perp W^\perp. \] + + Pertanto ogni vettore $\v \in V$ può scriversi come $\w + \w'$ dove $\w \in W$ e $\w' \in W^\perp$, + dove $\varphi(\w, \w') = 0$. + \end{remark} + + \begin{definition} (proiezione ortogonale) + Si definisce l'applicazione $\pr_W : V \to V$, detta \textbf{proiezione ortogonale} su $W$, + in modo tale che $\pr_W(\v) = \w$, dove $\v = \w + \w'$, con $\w \in W$ e $\w' \in W^\perp$. + \end{definition} + + \begin{remark}\nl + \li Dacché la proiezione ortogonale è un caso particolare della classica applicazione lineare + di proiezione su un sottospazio di una somma diretta, $\pr_W$ è un'applicazione lineare. \\ + \li Vale chiaramente che $\pr_W^2 = \pr_W$, da cui si ricava, se $W^\perp \neq \zerovecset$, che + $\varphi_{\pr_W}(\lambda) = \lambda (\lambda -1)$, ossia che $\Sp(\pr_W) = \{0, 1\}$. Infatti + $\pr_W(\v)$ appartiene già a $W$, ed essendo la scrittura in somma di due elementi, uno di $W$ e + uno di $W'$, unica, $\pr_W(\pr_W(\v)) = \pr_W(\v)$, da cui l'identità $\pr_W^2 = \pr_W$. \\ + \li Seguendo il ragionamento di prima, vale anche che $\restr{\pr_W}{W} = \Idw$ e che + $\restr{\pr_W}{W^\perp} = 0$. \\ + \li Inoltre, vale la seguente riscrittura di $\v \in V$: $\v = \pr_W(\v) + \pr_{W^\perp}(\v)$. \\ + \li Se $\basis = \{ \vv1, \ldots, \vv n \}$ è una base ortogonale di $W$, allora + $\pr_W(\v) = \sum_{i=1}^n \frac{\varphi(\v, \vv i)}{\varphi(\vv i, \vv i)} \vv i = \sum_{i=1}^n C(\v, \vv i) \vv i$. Infatti $\v -\sum_{i=1}^n C(\v, \vv i) \vv i \in W^\perp$. \\ + \li $\pr_W$ è un operatore simmetrico (o hermitiano se lo spazio è complesso). Infatti $\varphi(\pr_W(\v), \w) = + \varphi(\pr_W(\v), \pr_W(\w) + \pr_{W^\perp}(\w)) = \varphi(\pr_W(\v), \pr_W(\w)) = \varphi(\pr_W(\v) + \pr_{W^\perp}(\v), \pr_W(\w)) = \varphi(\v, \pr_W(\w))$. + \end{remark} + + \begin{proposition} + Sia $(V, \varphi)$ uno spazio euclideo. Allora valgono i seguenti risultati: + + \begin{enumerate}[(i)] + \item Siano $U$, $W \subseteq V$ sono sottospazi di $V$, allora $U \perp W$, ossia\footnote{È sufficiente che valga $U \subseteq W^\perp$ affinché valga anche $W \subseteq U^\perp$. Infatti $U \subseteq W^\perp \implies W = W^\dperp \subseteq U^\perp$. Si osserva che in generale vale che $W \subseteq W^\dperp$, dove vale l'uguaglianza nel caso di un prodotto $\varphi$ non degenere, com'è nel caso di uno spazio euclideo, + essendo $\varphi > 0$ per ipotesi.} $U \subseteq W^\perp$, $\iff \pr_U \circ \pr_W = \pr_W \circ \pr_U = 0$. + + \item Sia $V = W_1 \oplus \cdots \oplus W_n$. Allora $\v = \sum_{i=1}^n \pr_{W_i}(\v)$ $\iff$ $W_i \perp W_j$ $\forall i \neq j$, $1 \leq i, j \leq n$. + \end{enumerate} + \end{proposition} + + \begin{proof} + Si dimostrano i due risultati separatamente. + + \begin{enumerate}[(i)] + \item Sia $\v \in V$. Allora $\pr_W(\v) \in W = W^\dperp \subseteq U^\perp$. Pertanto + $\pr_U(\pr_W(\v)) = \vec 0$. Analogamente $\pr_W(\pr_U(\v)) = \vec 0$, da cui la tesi. + + \item Sia vero che $\v = \sum_{i=1}^n \pr_{W_i}(\v)$ $\forall \v \in V$. Sia $\w \in W_j$. Allora $\w = \sum_{i=1}^n \pr_{W_i}(\w) = \w + \sum_{\substack{i=1 \\ i \neq j}} \pr_{W_i}(\w) \implies \pr_{W_i}(\w) = \vec 0$ $\forall i \neq j$. Quindi $\w \in W_i^\perp$ $\forall i \neq j$, e si conclude che $W_i \subseteq W_j^\perp + \implies W_i \perp W_j$. Se invece $W_i \perp W_j$ $\forall i \neq j$, sia $\basis_i = \left\{ \w_i^{(1)}, \ldots, \w_i^{(k_i)} \right\}$ una base ortogonale di $W_i$. Allora $\basis = \cup_{i=1}^n \basis_i$ è anch'essa + una base ortogonale di $V$, essendo $\varphi\left(\w_i^{(t_i)}, \w_j^{(t_j)}\right) = 0$ per ipotesi. + Pertanto $\v = \sum_{i=1}^n \sum_{j=1}^{k_i} C\left(\v, \w_i^{(j)}\right) \w_i^{(j)} = \sum_{i=1}^n \pr_{W_i}(\v)$, + da cui la tesi. \qedhere + \end{enumerate} + \end{proof} + + \begin{definition} (inversione ortogonale) + Si definisce l'applicazione $\rho_W : V \to V$, detta \textbf{inversione ortogonale}, in modo tale che, detto $\v = \w + \w' \in V$ con $\w \in W$, $\w \in W^\perp$, $\rho_W(\v) = \w - \w'$. Se $\dim W = \dim V - 1$, + si dice che $\rho_W$ è una \textbf{riflessione}. + \end{definition} + + \begin{remark}\nl + \li Si osserva che $\rho_W$ è un'applicazione lineare. \\ + \li Vale l'identità $\rho_W^2 = \Idv$, da cui si ricava che $\varphi_{\rho_W}(\lambda) \mid (\lambda-1)(\lambda+1)$. In particolare, se $W^\perp \neq \zerovecset$, vale proprio + che $\Sp(\rho_W) = \{\pm1\}$, dove $V_1 = W$ e $V_{-1} = W^\perp$. \\ + \li $\rho_W$ è ortogonale (o unitaria, se $V$ è uno spazio euclideo complesso). Infatti se $\vv 1 = \ww 1 + \ww 1'$ e $\vv 2 = \ww 2 + \ww 2 '$, con $\ww 1$, $\ww 2 \in W$ e $\ww 1'$, $\ww 2' \in W$, $\varphi(\rho_W(\vv 1), \rho_W(\vv 2)) = \varphi(\ww 1 - \ww 1', \ww 2 - \ww 2') = \varphi(\ww 1, \ww 2) \underbrace{- \varphi(\ww 1', \ww 2) - \varphi(\ww 1, \ww 2')}_{=\,0} + \varphi(\ww 1', \ww 2') = \varphi(\ww 1 - \ww 1', \ww 2 - \ww 2')$. \\ + + Quindi $\varphi(\rho_W(\vv 1), \rho_W(\vv 2)) = \varphi(\ww 1, \ww 2) + \varphi(\ww 1', \ww 2) + \varphi(\ww 1, \ww 2') + \varphi(\ww 1', \ww 2') = \varphi(\vv 1, \vv 2)$. + \end{remark} \end{document} \ No newline at end of file diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index b060951..b93bed5 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -110,7 +110,9 @@ \newcommand{\bigzero}{\mbox{0}} \newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} -\newcommand{\Idv}{Id_V} +\newcommand{\Idv}{\operatorname{Id}_V} +\newcommand{\Idw}{\operatorname{Id}_W} +\newcommand{\IdV}[1]{\operatorname{Id}_{#1}} \DeclareMathOperator{\CI}{CI} \DeclareMathOperator{\Bil}{Bil} \DeclareMathOperator{\Mult}{Mult} @@ -118,6 +120,10 @@ \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\Cof}{Cof} +\DeclareMathOperator{\pr}{pr} +\DeclareMathOperator{\Sp}{sp} + +\newcommand{\dperp}{{\perp\perp}} \newcommand{\Eigsp}[0]{V_{\lambda}} \newcommand{\Gensp}[0]{\widetilde{V_{\lambda}}}