From 2c5d3e3443c35ad9dc8ca228562eb83d9ff6c5c1 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 5 May 2023 00:07:18 +0200 Subject: [PATCH] feat(geometria): aggiunge le condizioni per la semidefinitezza di un prodotto scalare --- .../main.pdf | Bin 237514 -> 237510 bytes .../main.tex | 10 +- .../main.pdf | Bin 437375 -> 437374 bytes .../main.tex | 4 +- .../1. 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In tal caso, + poiché $\varphi$ non è definito, deve anche esistere $\U \in V$, $\U \neq \vec 0 \mid q(\U) = 0 \implies \CI(\varphi) \neq \zerovecset$. \\ + + Se invece tali $\v$, $\w$ esistono, questi sono anche linearmente indipendenti. Se infatti + non lo fossero, uno sarebbe il multiplo dell'altro, e quindi le loro due forme quadratiche + sarebbero concordi di segno, \Lightning. Si consideri allora la combinazione lineare + $\v + \lambda \w$ al variare di $\lambda \in \RR$, imponendo che essa sia isotropa: + + \[ q(\v + \lambda \w) = 0 \iff \lambda^2 q(\w)+ 2 \lambda q(\v, \w) + q(\v) = 0. \] + + \vskip 0.05in + + Dal momento che $\frac{\Delta}{4} = \overbrace{q(\v, \w)^2}^{\geq 0} - \overbrace{q(\w)q(\v)}^{> 0}$ è + sicuramente maggiore di zero, tale equazione ammette due soluzioni reali $\lambda_1$, $\lambda_2$. + In particolare $\lambda_1$ è tale che $\v + \lambda_1 \w \neq \vec 0$, dal momento che $\v$ e $\w$ + sono linearmente indipendenti. Allora $\v + \lambda_1 \w$ è un vettore isotropo non nullo + di $V$ $\implies \CI(\varphi) \neq \zerovecset$. \\ + + Si conclude allora, tramite la contronominale, che se $\CI(\varphi) = \zerovecset$, $\varphi$ + è necessariamente definito. +\end{proof} + +\begin{proposition} Sia $\KK = \RR$. Allora + $\varphi$ è semidefinito $\iff$ $\CI(\varphi) = V^\perp$. \label{prop:semidefinitezza_varphi} +\end{proposition} + +\begin{proof} + Si dimostrano le due implicazioni separatamente. \\ + + \rightproof Sia $\varphi$ semidefinito. Chiaramente $V^\perp \subseteq \CI(\varphi)$. Si assuma + che $V^\perp \subsetneq \CI(\varphi)$. Sia allora $\v$ tale che $\v \in \CI(\varphi)$ e che $\v \notin V^\perp$. + Poiché $\v \notin V^\perp$, esiste un vettore $\w \in V$ tale che $\varphi(\v, \w) \neq 0$. Si osserva + che $\v$ e $\w$ sono linearmente indipendenti tra loro. Se infatti non lo fossero, esisterebbe $\mu \in \RR$ + tale che $\w = \mu \v \implies \varphi(\v, \w) = \mu \, \varphi(\v, \v) = 0$, \Lightning. \\ + + Si consideri allora la combinazione lineare $\v + \lambda \w$. Si consideri $\varphi$ semidefinito positivo. + In tal caso si può imporre che la valutazione di $q$ in $\v + \lambda \w$ sia strettamente negativa: + + \[ q(\v + \lambda \w) < 0 \iff \overbrace{q(\v)}^{=0} + \lambda^2 q(\w) + 2 \lambda \, \varphi(\v, \w) < 0. \] + + \vskip 0.05in + + In particolare, dal momento che $\frac{\Delta}{4} = \varphi(\v, \w)^2 > 0$, tale disequazione ammette + una soluzione $\lambda_1 \neq 0$. Inoltre $\v + \lambda_1 \w \neq \vec 0$, dal momento che $\v$ e + $\w$ sono linearmente indipendenti. Allora si è trovato un vettore non nullo per cui la valutazione in esso + di $q$ è negativa, contraddicendo l'ipotesi di semidefinitezza positiva di $\varphi$, \Lightning. Analogamente + si dimostra la tesi per $\varphi$ semidefinito negativo. \\ + + \leftproof Sia $\varphi$ non semidefinito. Allora devono esistere $\v$, $\w \in V$ tali che + $q(\v) > 0$ e che $q(\w) < 0$. In particolare, $\v$ e $\w$ sono linearmente indipendenti + tra loro, dal momento che se non lo fossero, uno sarebbe multiplo dell'altro, e le + valutazioni in essi di $q$ sarebbero concordi di segno, \Lightning. Si consideri allora + la combinazione lineare $\v + \lambda \w$, imponendo che $q$ si annulli in essa: + + \[ q(\v + \lambda \w) = 0 \iff \lambda^2 q(\w)+ 2 \lambda q(\v, \w) + q(\v) = 0. \] + + \vskip 0.05in + + In particolare, dal momento che $\frac{\Delta}{4} = \varphi(\v, \w)^2 > 0$, tale disequazione ammette + una soluzione $\lambda_1 \neq 0$. Allora, per tale $\lambda_1$, $\v + \lambda_1 \w \in \CI(\varphi)$. + Tuttavia $\varphi(\v + \lambda_1 \w, \v - \lambda_1 \w) = q(\v) - \underbrace{\lambda_1^2 q(\w)}_{<0} > 0 \implies + \v + \lambda_1 \w \notin V^\perp \implies \CI(\varphi) \supsetneq V^\perp$. \\ + + Si conclude allora, tramite la contronominale, che se $\CI(\varphi) = V^\perp$, $\varphi$ + è necessariamente semidefinito. +\end{proof} + +\section{Formula delle dimensioni e di polarizzazione rispetto a $\varphi$} + +\begin{definition}[sottospazio ortogonale a $W$] + Sia $W \subseteq V$ un sottospazio di $V$. Si identifica allora come \textbf{sottospazio ortogonale a $W$} + il sottospazio $W^\perp = \{ \v \in V \mid \varphi(\v, \w) \, \forall \w \in W \}$. +\end{definition} + +\begin{proposition}[formula delle dimensioni del prodotto scalare] + Sia $W \subseteq V$ un sottospazio di $V$. Allora vale la seguente identità: + + \[ \dim W + \dim W^\perp = \dim V + \dim (W \cap V^\perp). \] +\end{proposition} + +\begin{proof} + Si consideri l'applicazione lineare $a_\varphi$ introdotta precedentemente. Si osserva che $W^\perp = \Ker (i^\top \circ a_\varphi)$, dove $i : W \to V$ è tale che $i(\vec w) = \vec w$. Allora, + per la formula delle dimensioni, vale la seguente identità: + + \begin{equation} + \label{eq:dim_formula_dimensioni_1} + \dim V = \dim W^\perp + \rg (i^\top \circ a_\varphi). + \end{equation} + + \vskip 0.05in + + Sia allora $f = i^\top \circ a_\varphi$. + Si consideri ora l'applicazione $g = a_\varphi \circ i : W \to \dual V$. Sia ora $\basis_W$ una base di $W$ e + $\basis_V$ una base di $V$. Allora le matrici associate di $f$ e di $g$ sono le seguenti: + + \begin{enumerate}[(i)] + \item $M_{\dual \basis_W}^{\basis_V}(f) = M_{\dual \basis_W}^{\basis_V}(i^\top \circ a_\varphi) = + \underbrace{M_{\dual \basis_W}^{\dual \basis_V}(i^\top)}_A \underbrace{M_{\dual \basis_V}^{\basis_V}(a_\varphi)}_B = AB$, + \item $M_{\dual \basis_V}^{\basis_W}(g) = M_{\dual \basis_V}^{\basis_W}(a_\varphi \circ i) = + \underbrace{M_{\dual \basis_V}^{\basis_V}(a_\varphi)}_B \underbrace{M_{\basis_V}^{\basis_W}(i)}_{A^\top} = BA^\top \overbrace{=}^{B^\top = B} (AB)^\top$. + \end{enumerate} + + Poiché $\rg(A) = \rg(A^\top)$, si deduce che $\rg(f) = \rg(g) \implies \rg(i^\top \circ a_\varphi) = \rg(a_\varphi \circ i) = \rg(\restr{a_\varphi}{W}) = \dim W - \dim \Ker \restr{a_\varphi}{W}$, ossia che: + + \begin{equation} + \label{eq:dim_formula_dimensioni_2} + \rg(i^\top \circ a_\varphi) = \dim W - \dim (W \cap \underbrace{\Ker a_\varphi}_{V^\perp}) = \dim W - \dim (W \cap V^\perp). + \end{equation} + + Si conclude allora, sostituendo l'equazione \eqref{eq:dim_formula_dimensioni_2} nell'equazione \eqref{eq:dim_formula_dimensioni_1}, che $\dim V = \dim W^\top + \dim W - \dim (W \cap V^\perp)$, ossia la tesi. +\end{proof} + +\begin{remark} Si identifica $\w^\perp$ come il sottospazio di tutti i vettori di $V$ ortogonali a $\w$. + In particolare, se $W = \Span(\vec w)$ è il sottospazio generato da $\vec w \neq \vec 0$, $\vec w \in V$, allora $W^\perp = \w^\perp$. Inoltre valgono le seguenti equivalenze: $\vec w \notin W^\perp \iff$ $\Rad (\restr{\varphi}{W}) = W \cap W^\perp = \zerovecset$ $\iff \vec w \text{ non è isotropo } \iff$ $V = W \oplus W^\perp$. +\end{remark} + +\begin{proposition}[formula di polarizzazione] + Se $\Char \KK \neq 2$, un prodotto scalare è univocamente determinato dalla sua forma quadratica $q$. + In particolare vale la seguente identità: + + \[ \varphi(\v, \w) = \frac{q(\v + \w) - q(\v) - q(\w)}{2}. \] + + \vskip 0.05in +\end{proposition} + +\section{Il teorema di Lagrange e basi ortogonali} + +\begin{definition} + Si definisce \textbf{base ortogonale} di $V$ una base $\vv 1$, ..., $\vv n$ tale per cui $\varphi(\vv i, \vv j) = 0 + \impliedby i \neq j$, ossia una base per cui la matrice associata del prodotto scalare è diagonale. +\end{definition} + +\begin{theorem}[di Lagrange] + Ogni spazio vettoriale $V$ su $\KK$ tale per cui $\Char \KK \neq 2$ ammette una base ortogonale. +\end{theorem} + +\begin{proof} + Si dimostra il teorema per induzione su $n := \dim V$. Per $n \leq 1$, la tesi è triviale (se esiste una base, tale base è + già ortogonale). Sia + allora il teorema vero per $i \leq n$. Se $V$ ammette un vettore non isotropo $\vec w$, sia $W = \Span(\vec w)$ e si consideri la decomposizione $V = W \oplus W^\perp$. Poiché $W^\perp$ ha dimensione $n-1$, per ipotesi induttiva + ammette una base ortogonale. Inoltre, tale base è anche ortogonale a $W$, e quindi l'aggiunta di $\vec w$ a + questa base ne fa una base ortogonale di $V$. Se invece $V$ non ammette vettori non isotropi, ogni forma quadratica + è nulla, e quindi il prodotto scalare è nullo per la proposizione precedente. Allora in questo caso + ogni base è una base ortogonale, completando il passo induttivo, e dunque la dimostrazione. +\end{proof} + +\subsection{L'algoritmo di ortogonalizzazione di Gram-Schmidt} + +\begin{definition}[coefficiente di Fourier] + Siano $\v \in V$ e $\w \in V \setminus \CI(\varphi)$. Allora si definisce il \textbf{coefficiente di Fourier} + di $\v$ rispetto a $\w$ come il rapporto $C(\w, \v) = \frac{\varphi(\v, \w)}{\varphi(\w, \w)}$. +\end{definition} + +Se $\CI(\varphi) = \zerovecset$ (e quindi nel caso di $\KK = \RR$, dalla +\textit{Proposizione \ref{prop:definitezza_varphi}}, se $\varphi$ è definito) ed è +data una base $\basis = \{ \vv 1, \ldots, \vv n \}$ per $V$, è possibile +applicare l'\textbf{algoritmo di ortogonalizzazione di Gram-Schmidt} per ottenere +da $\basis$ una nuova base $\basis' = \{ \vv 1', \ldots, \vv n' \}$ con le seguenti proprietà: + +\begin{enumerate}[(i)] + \item $\basis'$ è una base ortogonale, + \item $\basis'$ mantiene la stessa bandiera di $\basis$ (ossia $\Span(\vv 1, \ldots, \vv i) = \Span(\vv 1', \ldots, \vv i')$ per ogni $1 \leq i \leq n$). +\end{enumerate} + +L'algoritmo si applica nel seguente modo: si prenda in considerazione $\vv 1$ e si sottragga ad ogni altro vettore +della base il vettore $C(\vv 1, \vv i) \vv 1 = \frac{\varphi(\vv 1, \vv i)}{\varphi(\vv 1, \vv 1)} \vv 1$, +rendendo ortogonale ogni altro vettore della base con $\vv 1$. Si sta quindi applicando la mappa +$\vv i \mapsto \vv i - \frac{\varphi(\vv 1, \vv i)}{\varphi(\vv 1, \vv 1)} \vv i = \vv i ^{(1)}$. +Si verifica infatti che $\vv 1$ e $\vv i ^{(1)}$ sono ortogonali per $2 \leq i \leq n$: + +\[ \varphi(\vv 1, \vv i^{(1)}) = \varphi(\vv 1, \vv i) - \varphi\left(\vv 1, \frac{\varphi(\vv 1, \vv i)}{\varphi(\vv 1, \vv 1)} \vv i\right) = \varphi(\vv 1, \vv i) - \varphi(\vv 1, \vv i) = 0. \] + +Poiché $\vv 1$ non è isotropo, si deduce che vale la decomposizione $V = \Span(\vv 1) \oplus \Span(\vv 1)^\perp$. +In particolare $\dim \Span(\vv 1)^\perp = n-1$: essendo allora i vettori $\vv 2 ^{(1)}, \ldots, \vv n ^{(1)}$ +linearmente indipendenti e appartenenti a $\Span(\vv 1)^\perp$, ne sono una base. Si conclude quindi +che vale la seguente decomposizione: + +\[ V = \Span(\vv 1) \oplus^\perp \Span(\vv 2 ^{(1)}, \ldots, \vv n ^{(1)}). \] + +\vskip 0.05in + +Si riapplica dunque l'algoritmo di Gram-Schmidt prendendo come spazio vettoriale lo spazio generato dai +vettori a cui si è applicato precedentemente l'algoritmo, ossia $V' = \Span(\vv 2 ^{(1)}, \ldots, \vv n ^{(1)})$, +fino a che non si ottiene $V' = \zerovecset$. 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 99be17190b22548c4fab9c0a947ecc3cccd6ec3a..46328f21e5ff0c3c2312123768edcd415c158d88 100644 GIT binary patch delta 197715 zcmZ^~Wn3If|K`o$1a}ECxa;6fa1SoQ-CctW7Tf}a!QI_GxVyUt2@nYGgyo$3KL6+J zv%9aly1HztrutLY_jmPn2hMRben0~t3Wx_{2eJVjO>Eu?3ch1iFn6$YvjXyRL2`ir z7Vw-X#`im|d(Ee9^DYr38A%*m6{J#3{Oy{4hnl_R)4`o-!>J0-Q{Kw-A{$Jnh|G

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