From ab05e879456ae2906a654de1a38f1d4b033bf482 Mon Sep 17 00:00:00 2001 From: Gabriel Antonio Videtta Date: Tue, 20 Jun 2023 16:11:18 +0200 Subject: [PATCH] feat(geometria): aggiunge una dimostrazione alternativa alla formula delle dimensioni del prod. scalare --- .../1. Introduzione al prodotto scalare.tex | 102 +++++ .../2. I prodotti hermitiani.tex | 320 +++++++++++++ ...eoremi principali sul prodotto scalare.tex | 430 ------------------ ...3. Spazi euclidei e teorema spettrale.tex} | 327 +------------ .../main.pdf | Bin 663430 -> 615279 bytes .../main.tex | 9 +- 6 files changed, 431 insertions(+), 757 deletions(-) create mode 100644 Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. I prodotti hermitiani.tex delete mode 100644 Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. Proprietà e teoremi principali sul prodotto scalare.tex rename Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/{3. Prodotti hermitiani, spazi euclidei e teorema spettrale.tex => 3. Spazi euclidei e teorema spettrale.tex} (80%) 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 fb80dcd..1749f4a 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 @@ -357,6 +357,33 @@ 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{proof}[Dimostrazione alternativa] + Si consideri nuovamente l'applicazione lineare $\alpha_\varphi$ introdotta + precedentemente. Si osserva innanzitutto che\footnote{$\alpha_\varphi\inv$ in questo caso non indica un'eventuale applicazione inversa di $\alpha_\varphi$, ma indica l'insieme delle eventuali controimmagini degli elementi su cui è applicata.} $W^\perp = \alpha_\varphi\inv(\Ann(W))$. Allora vale la seguente identità: + + \begin{equation} + \label{eq:formula_dimensioni_dimostrazione_alternativa_1} + \alpha_\varphi(W^\perp) = \Ann(W) \cap \Im \alpha_\varphi. + \end{equation} + + \vskip 0.05in + + Si mostra che $\Im \alpha_\varphi = \Ann(V^\perp)$. Chiaramente $\Im \alpha_\varphi \subseteq \Ann(V^\perp)$: siano infatti $\v \in V$ e $\w \in V^\perp$, allora $\alpha_\varphi(\v)(\w) = \varphi(\v, \w) = 0$. Inoltre $\dim \Im \alpha_\varphi = \rg \alpha_\varphi = n - \dim \Ker \alpha_\varphi = \dim V - \dim V^\perp = \dim \Ann(V^\perp)$, + da cui segue l'uguaglianza dei due sottospazi. Allora l'equazione \eqref{eq:formula_dimensioni_dimostrazione_alternativa_1} si può riscrivere\footnote{Si è utilizzata l'identità $\Ann(U) \cap \Ann(W) = \Ann(U + W)$, dove $U$ e $W$ sono due sottospazi di $V$, nonché che $\Ker \alpha_\varphi = V^\perp$.} come: + + \[ \alpha_\varphi(W^\perp) = \Ann(W) \cap \Ann(V^\perp) = \Ann(W + V^\perp) \] + + da cui segue che: + + \[ \dim W^\perp - \dim (V^\perp \cap W^\perp) = \dim V - \dim (W + V^\perp), \] + + e quindi, applicando la formula di Grassmann, che\footnote{Ricordiamo che $V^\perp \subseteq W^\perp$ per ogni sottospazio $W$ di $V$, e quindi che $\dim (V^\perp \cap W^\perp) = \dim V^\perp$.}: + + \[ \dim W^\perp - \dim V^\perp = \dim V - \dim W - \dim V^\perp + \dim (W \cap V^\perp), \] + + 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: $\vec w \notin W^\perp \iff$ $\Rad (\restr{\varphi}{W}) = W \cap W^\perp = \zerovecset$ $\iff \vec w \text{ non è isotropo } \iff$ $V = W \oplus^\perp W^\perp$. \\ @@ -834,4 +861,79 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni $\varphi(\vv i ', \vv j ') = 0$, 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} + +\section{Isometrie tra spazi vettoriali} + +\begin{definition} (isometria) + Dati due spazi vettoriali $(V, \varphi)$ e + $(V', \varphi')$ dotati di prodotto scalare sullo stesso campo $\KK$, si dice che + $V$ e $V'$ sono \textbf{isometrici} se esiste un isomorfismo + $f$, detto \textit{isometria}, che preserva tali che prodotti, ossia tale che: + + \[ \varphi(\vec v, \vec w) = \varphi'(f(\vec v), f(\vec w)). \] +\end{definition} + +\begin{exercise} Sia $f : V \to V'$ un isomorfismo. Allora $f$ è un'isometria $\iff$ $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$ $\iff$ $\exists$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$ tale che $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. +\end{exercise} + +\begin{solution} Se $f$ è un'isometria, detta $\basis$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$ + dal momento che $f$ è anche un isomorfismo. Inoltre, dacché $f$ è un'isometria, vale sicuramente che + $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. \\ + + Sia ora assunto per ipotesi che $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Allora, analogamente a prima, detta $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$, e in quanto tale, + per ipotesi, è tale che $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. \\ + + Sia infine assunto per ipotesi che $\exists$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$ tale che $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Siano $\v$, $\w \in V$. Allora $\exists a_1$, ..., $a_n$, $b_1$, ..., $b_n \in \KK$ + tali che $\v = a_1 \vv 1 + \ldots + a_n \vv n$ e $\w = b_1 \vv 1 + \ldots + b_n \vv n$. Si ricava pertanto + che: + + \[ \varphi'(f(\v), f(\w)) = \sum_{i=1}^n \sum_{j=1}^n a_i b_j \, \varphi'(f(\vv i), f(\vv j)) = + \sum_{i=1}^n \sum_{j=1}^n a_i b_j \, \varphi(\vv i, \vv j) = \varphi(\v, \w), \] + + da cui la tesi. +\end{solution} + +\begin{proposition} Sono equivalenti le seguenti affermazioni: + + \begin{enumerate}[(i)] + \item $V$ e $V'$ sono isometrici; + \item $\forall$ base $\basis$ di $V$, base $\basis'$ di $V'$, + $M_\basis(\varphi)$ e $M_{\basis'}(\varphi')$ sono congruenti; + \item $\exists$ base $\basis$ di $V$, base $\basis'$ di $V'$ tale che + $M_\basis(\varphi)$ e $M_{\basis'}(\varphi')$ sono congruenti. + \end{enumerate} +\end{proposition} + +\begin{proof} Se $V$ e $V'$ sono isometrici, sia $f : V \to V'$ un'isometria. Sia $\basisC = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Allora, poiché $f$ è anche un isomorfismo, $\basisC' = f(\basisC)$ è una base di $V$ tale che + $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Pertanto $M_\basisC(\varphi) = M_{\basisC'}(\varphi')$. Si conclude allora che, cambiando base in $V$ (o in $V'$), la matrice associata + al prodotto scalare varia per congruenza dalla formula di cambiamento di base per il prodotto scalare, da cui si ricava che per ogni scelta di $\basis$ base di $V$ e di $\basis'$ base di $V'$, $M_\basis(\varphi) \cong M_{\basis'}(\varphi')$. Inoltre, se tale risultato è vero per ogni $\basis$ base di $V$ e di $\basis'$ base di $V'$, dal momento che sicuramente esistono due basi $\basis$, $\basis'$ di $V$ e $V'$, vale anche (ii) $\implies$ (iii). \\ + + Si dimostra ora (iii) $\implies$ (i). Per ipotesi $M_\basis(\varphi) \cong M_{\basis'}(\varphi')$, quindi + $\exists P \in \GL(n, \KK) \mid M_{\basis'}(\varphi') = P^\top M_\basis(\varphi) P$. Allora $\exists$ $\basis''$ + base di $V'$ tale che $P = M_{\basis''}^{\basis'}(\Idv)$, da cui $P\inv = M_{\basis'}^{\basis''}(\varphi)$. Per la formula di cambiamento di base del prodotto + scalare, $M_{\basis''}(\varphi) = (P\inv)^\top M_{\basis'} P\inv = M_\basis(\varphi)$. Detta + $\basis'' = \{ \ww 1, \ldots, \ww n \}$, si costruisce allora l'isomorfismo $f : V \to V'$ tale + che $f(\vv i) = \ww i$ $\forall 1 \leq i \leq n$.. Dal momento che per costruzione $M_\basis(\varphi) = M_{\basis''}(\varphi')$, + $\varphi(\vv i, \vv j) = \varphi'(\ww i, \ww j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. + Si conclude dunque che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v, \w \in V$, e dunque + che $f$ è un'isometria, come desiderato dalla tesi. +\end{proof} + +\begin{proposition} $(V, \varphi)$ e $(V', \varphi')$ spazi vettoriali + su $\RR$ sono + isometrici $\iff$ $\varphi$ e $\varphi'$ hanno la stessa segnatura. +\end{proposition} + +\begin{proof}\nl\nl + \rightproof Per la precedente proposizione, esistono due basi $\basis$ e $\basis'$, una di $V$ e una di $V'$, + tali che $M_\basis(\varphi) \cong M_{\basis'}(\varphi)$. Allora queste due matrici condividono la stessa + segnatura, e così quindi anche $\varphi$ e $\varphi'$. \\ + + \leftproof Se $\varphi$ e $\varphi'$ hanno la stessa segnatura, esistono due basi $\basis = \{ \vv 1, \ldots, \vv n \}$ e $\basis' = \{ \ww 1, \ldots, \ww n \}$, una + di $V$ e una di $V'$, tali che $M = M_\basis(\varphi) = M_{\basis'}(\varphi')$ e che $M$ è una matrice di + Sylvester. Allora si costruisce $f : V \to V'$ tale che $f(\vv i) = \ww i$. Esso è un isomorfismo, e per + costruzione $\varphi(\vv i, \vv j) = \varphi'(\ww i, \ww j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$, da cui + si conclude che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v$, $\w \in V$, e quindi che $V$ e + $V'$ sono isometrici. \end{proof} \ No newline at end of file diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. I prodotti hermitiani.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. I prodotti hermitiani.tex new file mode 100644 index 0000000..17ad40e --- /dev/null +++ b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. I prodotti hermitiani.tex @@ -0,0 +1,320 @@ +\chapter{I prodotti hermitiani e complessificazione (non indicizzato)} + +\begin{definition} (prodotto hermitiano) Sia $\KK = \CC$. Una mappa $\varphi : V \times V \to \CC$ si dice \textbf{prodotto hermitiano} se: + + \begin{enumerate}[(i)] + \item $\varphi$ è $\CC$-lineare nel secondo argomento, ossia se $\varphi(\v, \U + \w) = \varphi(\v, \U) + \varphi(\v, \w)$ e + $\varphi(\v, a \w) = a \, \varphi(\v, \w)$, + \item $\varphi(\U, \w) = \conj{\varphi(\w, \U)}$. + \end{enumerate} +\end{definition} + +\begin{definition} (prodotto hermitiano canonico in $\CC^n$) Si definisce + \textbf{prodotto hermitiano canonico} di $\CC^n$ il prodotto $\varphi : \CC^n \times \CC^n \to \CC$ tale per cui, detti $\v = (z_1 \cdots z_n)^\top$ e $\w = (w_1 \cdots w_n)^\top$, $\varphi(\v, \w) = \sum_{i=1}^n \conj{z_i} w_i$. +\end{definition} + +\begin{remark}\nl + \li $\varphi(\U + \w, \v) = \conj{\varphi(\v, \U + \w)} = + \conj{\varphi(\v, \U) + \varphi(\v, \w)} = \conj{\varphi(\v, \U)} + \conj{\varphi(\v, \U)} = \varphi(\w, \v) + \varphi(\U, \v)$, ossia + $\varphi$ è additiva anche nel primo argomento. \\ + \li $\varphi(a \v, \w) = \conj{\varphi(\w, a \v)} = \conj{a} \conj{\varphi(\w, \v)} = \conj{a} \, \varphi(\v, \w)$. \\ + \li $\varphi(\v, \v) = \conj{\varphi(\v, \v)}$, e quindi $\varphi(\v, \v) \in \RR$. \\ + \li Sia $\v = \sum_{i=1}^n x_i \vv i$ e sia $\w = \sum_{i=1}^n y_i \vv i$, allora $\varphi(\v, \w) = \sum_{i =1}^n \sum_{j=1}^n \conj{x_i} y_i \varphi(\vv i, \vv j)$. \\ + \li $\varphi(\v, \w) = 0 \iff \varphi(\w, \v) = 0$. +\end{remark} + +\begin{proposition} + Data la forma quadratica $q : V \to \RR$ del prodotto hermitiano $\varphi$ tale che $q(\v) = \varphi(\v, \v) \in \RR$, tale + forma quadratica individua univocamente il prodotto hermitiano $\varphi$. +\end{proposition} + +\begin{proof} + Innanzitutto si osserva che: + + \[ \varphi(\v, \w) = \frac{\varphi(\v, \w) + \conj{\varphi(\v, \w)}}{2} + \frac{\varphi(\v, \w) . \conj{\varphi(\v, \w)}}{2}. \] + + \vskip 0.05in + + Si considerano allora le due identità: + + \[ q(\v + \w) - q(\v) - q(\w) = + \varphi(\v, \w) + \conj{\varphi(\w, \v)} = 2 \, \Re(\varphi(\v, \w)), \] + + \[ q(i\v + \w) - q(\v) - q(\w) = -i(\varphi(\v, \w) - \conj{\varphi(\v, \w)}) = 2 \, \imm(\varphi(\v, \w)), \] + + \vskip 0.05in + + da cui si conclude che il prodotto $\varphi$ è univocamente + determinato dalla sua forma quadratica. +\end{proof} + +\begin{definition} + Si definisce \textbf{matrice aggiunta} di $A \in M(n, \KK)$ la matrice coniugata della trasposta di $A$, ossia: + + \[ A^* = \conj{A^\top} = \conj{A}^\top. \] +\end{definition} + +\begin{remark} + Per quanto riguarda la matrice aggiunta valgono le principali proprietà della matrice trasposta: + + \begin{itemize} + \item $(A + B)^* = A^* + B^*$, + \item $(AB)^* = B^* A^*$, + \item $(A\inv)^* = (A^*)\inv$, se $A$ è invertibile. + \end{itemize} +\end{remark} + +%TODO: aggiungere tr(conj(A^t) B) + +\begin{definition} (matrice associata del prodotto hermitiano) Analogamente + al caso del prodotto scalare, data una base $\basis = \{\vv 1, \ldots, \vv n\}$ si definisce + come \textbf{matrice associata del prodotto hermitiano} $\varphi$ + la matrice $M_\basis(\varphi) = (\varphi(\vv i, \vv j))_{i,j = 1 \textrm{---} n}$. +\end{definition} + +\begin{remark} + Si osserva che, analogamente al caso del prodotto scalare, vale + la seguente identità: + + \[ \varphi(\v, \w) = [\v]_\basis^* M_\basis(\varphi) [\w]_\basis. \] +\end{remark} + +\begin{proposition} + (formula del cambiamento di base per i prodotto hermitiani) Siano + $\basis$, $\basis'$ due basi di $V$. Allora vale la seguente + identità: + + \[ M_{\basis'} = M_{\basis}^{\basis'}(\Idv)^* M_\basis(\varphi) M_{\basis}^{\basis'}(\Idv). \] +\end{proposition} + +\begin{proof} + Siano $\basis = \{ \vv 1, \ldots, \vv n \}$ e $\basis' = \{ \ww 1, \ldots, \ww n \}$. Allora $\varphi(\ww i, \ww j) = [\ww i]_\basis^* M_\basis(\varphi) [\ww j]_\basis = \left( M_\basis^{\basis'}(\Idv)^i \right)^* M_\basis(\varphi) M_\basis^{\basis'}(\Idv)^j = + \left(M_\basis^{\basis'}(\Idv)\right)^*_i M_\basis(\varphi) M_\basis^{\basis'}(\Idv)^j$, da cui si ricava l'identità + desiderata. +\end{proof} + +\begin{definition} (radicale di un prodotto hermitiano) + Analogamente al caso del prodotto scalare, si definisce il \textbf{radicale} del prodotto $\varphi$ come il seguente sottospazio: + + \[ V^\perp = \{ \v \in V \mid \varphi(\v, \w) = 0 \, \forall \w \in V \}. \] +\end{definition} + +\begin{proposition} + Sia $\basis$ una base di $V$ e $\varphi$ un prodotto hermitiano. Allora $V^\perp = [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$\footnote{Stavolta non è sufficiente considerare la mappa $f : V \to V^*$ tale che $f(\v) = \left[ \w \mapsto \varphi(\v, \w) \right]$, dal momento che $f$ non è lineare, bensì antilineare, ossia $f(a \v) = \conj a f(\v)$.}. +\end{proposition} + +\begin{proof} + Sia $\basis = \{ \vv 1, \ldots, \vv n \}$ e sia $\v \in V^\perp$. + Siano $a_1$, ..., $a_n \in \KK$ tali che $\v = a_1 \vv 1 + \ldots + a_n \vv n$. Allora, poiché $\v \in V$, $0 = \varphi(\vv i, \v) + = a_1 \varphi(\vv i, \vv 1) + \ldots + a_n \varphi(\vv i, \vv n) = M_i [\v]_\basis$, da cui si ricava che $[\v]_\basis \in \Ker M_\basis(\varphi)$, e quindi che $V^\perp \subseteq [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$. \\ + + Sia ora $\v \in V$ tale che $[\v]_\basis \in \Ker M_\basis(\varphi)$. + Allora, per ogni $\w \in V$, $\varphi(\w, \v) = [\w]_\basis^* M_\basis(\varphi) [\v]_\basis = [\w]_\basis^* 0 = 0$, da cui si + conclude che $\v \in V^\perp$, e quindi che $V^\perp \supseteq [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$, da cui + $V^\perp = [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$, ossia + la tesi. +\end{proof} + +\begin{remark} + Come conseguenza della proposizione appena dimostrata, valgono + le principali proprietà già viste per il prodotto scalare. \\ + + \li $\det(M_\basis(\varphi)) = 0 \iff V^\perp \neq \zerovecset \iff \varphi$ è degenere, \\ + \li Vale il teorema di Lagrange, e quindi quello di Sylvester, benché con alcune accortezze: si + introduce, come nel caso di $\RR$, il concetto di segnatura, che diventa l'invariante completo + della nuova congruenza hermitiana, che ancora una volta si dimostra essere una relazione + di equivalenza. \\ + \li Come mostrato nei momenti finali del documento (vd.~\textit{Esercizio 3}), vale + la formula delle dimensioni anche nel caso del prodotto hermitiano. +\end{remark} + +\hr + +\begin{definition} (restrizione ai reali di uno spazio) Sia $V$ + uno spazio vettoriale su $\CC$ con base $\basis$. Si definisce allora lo spazio $V_\RR$, detto + \textbf{spazio di restrizione su $\RR$} di $V$, come uno spazio su $\RR$ generato da + $\basis_\RR = \basis \cup i \basis$. +\end{definition} + +\begin{example} + Si consideri $V = \CC^3$. Una base di $\CC^3$ è chiaramente $\{ \e1, \e2, \e3 \}$. Allora + $V_\RR$ sarà uno spazio vettoriale su $\RR$ generato dai vettori $\{ \e1, \e2, \e3, i\e1, i\e2, i\e3 \}$. +\end{example} + +\begin{remark} + Si osserva che lo spazio di restrizione su $\RR$ e lo spazio di partenza condividono lo stesso insieme + di vettori. Infatti, $\Span_\CC(\basis) = \Span_\RR(\basis \cup i\basis)$. Ciononostante, $\dim V_\RR = 2 \dim V$\footnote{Si sarebbe potuto ottenere lo stesso risultato utilizzando il teorema delle torri algebriche: $[V_\RR : \RR] = [V: \CC] [\CC: \RR] = 2 [V : \CC]$.}, se $\dim V \in \NN$. +\end{remark} + +\begin{definition} (complessificazione di uno spazio) Sia $V$ uno spazio vettoriale su $\RR$. + Si definisce allora lo \textbf{spazio complessificato} $V_\CC = V \times V$ su $\CC$ con le seguenti operazioni: + + \begin{itemize} + \item $(\v, \w) + (\v', \w') = (\v + \v', \w + \w')$, + \item $(a+bi)(\v, \w) = (a\v - b\w, a\w + b\v)$. + \end{itemize} +\end{definition} + +\begin{remark} + La costruzione dello spazio complessificato emula in realtà la costruzione di $\CC$ come spazio + $\RR \times \RR$. Infatti se $z = (c, d)$, vale che $(a + bi)(c, d) = (ac - bd, ad + bc)$, mentre + si mantiene l'usuale operazione di addizione. In particolare si può identificare l'insieme + $V \times \zerovecset$ come $V$, mentre $\zerovecset \times V$ viene identificato come l'insieme + degli immaginari $iV$ di $V_\CC$. Infine, moltiplicare per uno scalare reale un elemento di + $V \times \zerovecset$ equivale a moltiplicare la sola prima componente con l'usuale operazione + di moltiplicazione di $V$. Allora, come accade per $\CC$, si può sostituire la notazione + $(\v, \w)$ con la più comoda notazione $\v + i \w$. +\end{remark} + +\begin{remark} + Sia $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Innanzitutto si osserva che + $(a+bi)(\v, \vec 0) = (a\v, b\v)$. Pertanto si può concludere che $\basis \times \zerovecset$ è + una base dello spazio complessificato $V_\CC$ su $\CC$. \\ + + Infatti, se $(a_1 + b_1 i)(\vv 1, \vec 0) + \ldots + (a_n + b_n i)(\vv n, \vec 0) = (\vec 0, \vec 0)$, + allora $(a_1 \vv 1 + \ldots + a_n \vv n, b_1 \vv 1 + \ldots + b_n \vv n) = (\vec 0, \vec 0)$. + Poiché però $\basis$ è linearmente indipendente per ipotesi, l'ultima identità implica che + $a_1 = \cdots = a_n = b_1 = \cdots = b_n = 0$, e quindi che $\basis \times \zerovecset$ è linearmente + indipendente. \\ + + Inoltre $\basis \times \zerovecset$ genera $V_\CC$. Se infatti $\v = (\U, \w)$, e vale che: + + \[ \U = a_1 \vv 1 + \ldots + a_n \vv n, \quad \w = b_1 \vv 1 + \ldots + b_n \vv n, \] + + \vskip 0.1in + + allora $\v = (a_1 + b_1 i) (\vv 1, \vec 0) + \ldots + (a_n + b_n i) (\vv n, \vec 0)$. Quindi + $\dim V_\CC = \dim V$. +\end{remark} + +\begin{definition} + Sia $f$ un'applicazione $\CC$-lineare di $V$ spazio vettoriale su $\CC$. Allora + si definisce la \textbf{restrizione su} $\RR$ di $f$, detta $f_\RR : V_\RR \to V_\RR$, + in modo tale che $f_\RR(\v) = f(\v)$. +\end{definition} + +\begin{remark} + Sia $\basis = \{\vv 1, \ldots, \vv n\}$ una base di $V$ su $\CC$. Sia $A = M_\basis(f)$. Si + osserva allora che, se $\basis' = \basis \cup i \basis$ e $A = A' + i A''$ con $A'$, $A'' \in M(n, \RR)$, + vale la seguente identità: + + \[ M_{\basis'}(f_\RR) = \Matrix{ A' & \rvline & -A'' \\ \hline A'' & \rvline & A' }. \] + + Infatti, se $f(\vv i) = (a_1 + b_1 i) \vv 1 + \ldots + (a_n + b_n i) \vv n$, vale che + $f_\RR(\vv i) = a_1 \vv 1 + \ldots + a_n \vv n + b_1 (i \vv 1) + \ldots + b_n (i \vv n)$, + mentre $f_\RR(i \vv i) = i f(\vv i) = - b_1 \vv 1 + \ldots - b_n \vv n + a_1 (i \vv 1) + \ldots + a_n (i \vv n)$. +\end{remark} + +\begin{definition} + Sia $f$ un'applicazione $\RR$-lineare di $V$ spazio vettoriale su $\RR$. Allora + si definisce la \textbf{complessificazione} di $f$, detta $f_\CC : V_\CC \to V_\CC$, + in modo tale che $f_\CC(\v + i \w) = f(\v) + i f(\w)$. +\end{definition} + +\begin{remark} + Si verifica infatti che $f_\CC$ è $\CC$-lineare. + \begin{itemize} + \item $f_\CC((\vv1 + i \ww1) + (\vv2 + i \ww2)) = f_\CC((\vv1 + \vv2) + i (\ww1 + \ww2)) = + f(\vv1 + \vv2) + i f(\ww1 + \ww2) = (f(\vv1) + i f(\ww1)) + (f(\vv2) + i f(\ww2)) = + f_\CC(\vv1 + i\ww1) + f_\CC(\vv2 + i\ww2)$. + + \item $f_\CC((a+bi)(\v + i\w)) = f_\CC(a\v-b\w + i(a\w+b\v)) = f(a\v - b\w) + i f(a\w + b\v) = + af(\v) - bf(\w) + i(af(\w) + bf(\v)) = (a+bi)(f(\v) + if(\w)) = (a+bi) f_\CC(\v + i\w)$. + \end{itemize} +\end{remark} + +\begin{proposition} + Sia $f_\CC$ la complessificazione di $f \in \End(V)$, dove $V$ è uno spazio vettoriale su $\RR$. + Sia inoltre $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Valgono allora i seguenti risultati: + + \begin{enumerate}[(i)] + \item $\restr{(f_\CC)_\RR}{V}$ assume gli stessi valori di $f$, + \item $M_\basis(f_\CC) = M_\basis(f) \in M(n, \RR)$, + \item $M_{\basis \cup i \basis}((f_\CC)_\RR) = \Matrix{M_\basis(f) & \rvline & 0 \\ \hline 0 & \rvline & M_\basis(f)}$. + \end{enumerate} +\end{proposition} + +\begin{proof}Si dimostrano i risultati separatamente. + \begin{enumerate}[(i)] + \item Si osserva che $(f_\CC)_\RR(\vv i) = f_\CC(\vv i) = f(\vv i)$. Dal momento che + $(f_\CC)_\RR$ è $\RR$-lineare, si conclude che $(f_\CC)_\RR$ assume gli stessi valori + di $f$. + + \item Dal momento che $\basis$, nell'identificazione di $(\v, \vec 0)$ come $\v$, è + sempre una base di $V_\CC$, e $f_\CC(\vv i) = f(\vv i)$, chiaramente + $[f_\CC(\vv i)]_\basis = [f(\vv i)]_\basis$, e quindi $M_\basis(f_\CC) = M_\basis(f)$, + dove si osserva anche che $M_\basis(f) \in M(n, \RR)$, essendo $V$ uno spazio vettoriale + su $\RR$. + + \item Sia $f(\vv i) = a_1 \vv 1 + \ldots + a_n \vv n$ con $a_1$, ..., $a_n \in \RR$. Come + osservato in (i), $\restr{(f_\CC)_\RR}{\basis} = \restr{(f_\CC)_\RR}{\basis}$, e quindi + la prima metà di $M_{\basis \cup i \basis}((f_\CC)_\RR)$ è formata da due blocchi: uno + verticale coincidente con $M_\basis(f)$ e un altro completamente nullo, dal momento che + non compare alcun termine di $i \basis$ nella scrittura di $(f_\CC)_\RR(\vv i)$. Al + contrario, per $i \basis$, $(f_\CC)_\RR(i \vv i) = f_\CC(i \vv i) = i f(\vv i) = a_1 (i \vv 1) + + \ldots + a_n (i \vv n)$; pertanto la seconda metà della matrice avrà i due blocchi della prima metà, + benché scambiati. + \end{enumerate} +\end{proof} + +\begin{remark} + Dal momento che $M_\basis(f_\CC) = M_\basis(f)$, $f_\CC$ e $f$ condividono lo stesso polinomio caratteristico + e vale che $\Sp(f) \subseteq \Sp(f_\CC)$, dove vale l'uguaglianza se e solo se tale polinomio caratteristico + è completamente riducibile in $\RR$. Inoltre, se $V_\lambda$ è l'autospazio su $V$ dell'autovalore $\lambda$, l'autospazio + su $V_\CC$, rispetto a $f_\CC$, è invece ${V_\CC}_\lambda = V_\lambda + i V_\lambda$, la cui + dimensione rimane invariata rispetto a $V_\lambda$, ossia $\dim V_\lambda = \dim {V_\CC}_\lambda$ + (infatti, analogamente a prima, una base di $V_\lambda$ può essere identificata come base + anche per ${V_\CC}_\lambda$). +\end{remark} + +\begin{proposition} + Sia $f_\CC$ la complessificazione di $f \in \End(V)$, dove $V$ è uno spazio vettoriale su $\RR$. + Sia inoltre $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Allora un endomorfismo + $\tilde g : V_\CC \to V_\CC$ complessifica un endomorfismo $g \in \End(V)$ $\iff$ $M_\basis(\tilde g) \in M(n, \RR)$. +\end{proposition} + +\begin{proof} + Se $\tilde g$ complessifica $g \in \End(V)$, allora, per la proposizione precedente, + $M_\basis(\tilde g) = M_\basis(g) \in M(n, \RR)$. Se invece $A = M_\basis(\tilde g) \in M(n, \RR)$, + si considera $g = M_\basis\inv(A) \in \End(V)$. Si verifica facilemente che $\tilde g$ non è altro che + il complessificato di tale $g$: + + \begin{itemize} + \item $\tilde g (\vv i) = g(\vv i)$, dove l'uguaglianza è data dal confronto delle matrici associate, + e quindi $\restr{\tilde g}{V} = g$; + \item $\tilde g(\v + i\w) = \tilde g(\v) + i \tilde g(\w) = g(\v) + i g(\w)$, da cui la tesi. + \end{itemize} +\end{proof} + +\begin{proposition} + Sia $\varphi$ un prodotto scalare di $V$ spazio vettoriale su $\RR$. Allora esiste un + unico prodotto hermitiano $\varphi_\CC : V_\CC \times V_\CC \to \CC$ che estende $\varphi$ (ossia tale che + $\restr{\varphi_\CC}{V \times V} = \varphi$), il quale assume la stessa segnatura + di $\varphi$. +\end{proposition} + +\begin{proof} + Sia $\basis$ una base di Sylvester per $\varphi$. Si consideri allora il prodotto + $\varphi_\CC$ tale che: + + \[ \varphi_\CC(\vv1 + i\ww1, \vv2 + i\ww2) = \varphi(\vv1, \vv2) + \varphi(\ww1, \ww2) + i(\varphi(\vv1, \ww1) - \varphi(\ww1, \vv2)). \] + + Chiaramente $\restr{\varphi_\CC}{V \times V} = \varphi$. Si verifica allora che $\varphi_\CC$ è hermitiano: + + \begin{itemize} + \item $\varphi_\CC(\v + i\w, (\vv1 + i\ww1) + (\vv2 + i\ww2))$ $= \varphi(\v, \vv1 + \vv2) + \varphi(\w, \ww1 + \ww2)$ $+ i(\varphi(\v, \ww1 + \ww2)$ $- \varphi(\w, \vv1 + \vv2))$ $= [\varphi(\v, \vv1) + \varphi(\w, \ww1) + i(\varphi(\v, \ww1) - \varphi(\w, \vv1))]$ $+ [\varphi(\v, \vv2) + \varphi(\w, \ww2) + i(\varphi(\v, \ww2) - \varphi(\w, \vv2))] = \varphi_\CC(\v + i\w, \vv1 + i\ww1) + + \varphi_\CC(\v + i\w, \vv2 + i\ww2)$ (additività nel secondo argomento), + + \item $\varphi_\CC(\v + i\w, (a+bi)(\vv1 + i\ww1)) = \varphi_\CC(\v + i\w, a\vv1-b\ww1 + i(b\vv1+a\ww1)) = + \varphi(\v, a\vv1-b\ww1) + \varphi(\w, b\vv1+a\ww1) + i(\varphi(\v, b\vv1+a\ww1) - \varphi(\w, a\vv1-b\ww1))= + a\varphi(\v, \vv1) - b\varphi(\v, \ww1) + b\varphi(\w, \vv1) + a\varphi(\w, \ww1) + i(b\varphi(\v, \vv1) + a\varphi(\v, \ww1) - a\varphi(\w, \vv1) + b\varphi(\w, \ww1)) = a(\varphi(\v, \vv1) + \varphi(\w, \ww1)) - + b(\varphi(\v, \ww1) - \varphi(\w, \vv1)) + i(a(\varphi(\v, \ww1) - \varphi(\w, \vv1)) + b(\varphi(\v, \vv1) + \varphi(\w, \ww1))) = (a+bi)(\varphi(\v, \vv1) + \varphi(\w, \ww1) + i(\varphi(\v, \ww1) - \varphi(\w, \vv1))) = (a+bi) \varphi_\CC(\v + \w, \vv1 + i\ww1)$ (omogeneità nel secondo argomento), + + \item $\varphi_\CC(\vv1 + i\ww1, \vv2 + i\ww2) = \varphi(\vv1, \vv2) + \varphi(\ww1, \ww2) + i(\varphi(\vv1, \ww2) - \varphi(\ww1, \vv2)) = \conj{\varphi(\vv1, \vv2) + \varphi(\ww1, \ww2) + i(\varphi(\ww1, \vv2) - \varphi(\vv1, \ww2))} = \conj{\varphi(\vv2, \vv1) + \varphi(\ww2, \ww1) + i(\varphi(\vv2, \ww1) - \varphi(\ww2, \vv1))} = \conj{\varphi_\CC(\vv2 + \ww2, \vv1 + \ww1)}$ (coniugio nello scambio degli argomenti). + \end{itemize} + + Ogni prodotto hermitiano $\tau$ che estende il prodotto scalare $\varphi$ ha la stessa matrice associata nella + base $\basis$, essendo $\tau(\vv i, \vv i) = \varphi(\vv i, \vv i)$ vero per ipotesi. Pertanto $\tau$ è + unico, e vale che $\tau = \varphi_\CC$. Dal momento che $M_\basis(\varphi_\CC) = M_\basis(\varphi)$ è + una matrice di Sylvester, $\varphi_\CC$ mantiene anche la stessa segnatura di $\varphi$. +\end{proof} diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. Proprietà e teoremi principali sul prodotto scalare.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. Proprietà e teoremi principali sul prodotto scalare.tex deleted file mode 100644 index f724bfd..0000000 --- a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/2. Proprietà e teoremi principali sul prodotto scalare.tex +++ /dev/null @@ -1,430 +0,0 @@ -\chapter{Proprietà e teoremi principali sul prodotto scalare} - -\begin{note} - Nel corso del documento, per $V$ si intenderà uno spazio vettoriale di dimensione - finita $n$ e per $\varphi$ un suo prodotto scalare. Analogamente si intenderà lo stesso - per $V'$ e $\varphi'$. -\end{note} - -\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 $f : V \to \dual W$ tale che $f(\vec v)$ è un funzionale di $\dual W$ tale che - $f(\vec v)(\vec w) = \varphi(\vec v, \vec w)$ $\forall \vec w \in W$. Si osserva che $W^\perp = \Ker f$, da cui, - per la formula delle dimensioni, $\dim V = \dim W^\perp + \rg f$. Inoltre, si osserva anche che - $f = i^\top \circ a_\varphi$, dove $i : W \to V$ è tale che $i(\vec w) = \vec w$, infatti $f(\vec v) = a_\varphi(\vec v) \circ i$ è un funzionale di $\dual W$ tale che $f(\vec v)(\vec w) = \varphi(\vec v, \vec w)$. Pertanto - $\rg f = \rg (i^\top \circ a_\varphi)$. \\ - - Si consideri ora l'applicazione $g = a_\varphi \circ i : W \to \dual W$. Sia ora $\basis_W$ una base di $W$ e - $\basis_V$ una base di $V$. Allora le matrice 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 $\rg(f) = \rg(g) \implies \rg(i^\top \circ a_\varphi) = \rg(a_\varphi \circ i) = \rg(\restr{a_\varphi}{W}) = \dim W - \dim \Ker \restr{a_\varphi}{W} = \dim W - \dim (W \cap \underbrace{\Ker a_\varphi}_{V^\perp}) = \dim W - \dim (W \cap V^\perp)$. Si conclude allora, sostituendo quest'ultima - identità nell'identità ricavata a inizio dimostrazione che $\dim V = \dim W^\top + \dim W - \dim (W \cap V^\perp)$, - ossia la tesi. -\end{proof} - -\begin{remark} - Si possono fare alcune osservazioni sul radicale di un solo elemento $\vec w$ e su quello del suo sottospazio - generato $W = \Span(\vec w)$: \\ - - \li $\vec w ^\perp = W^\perp$, \\ - \li $\vec w \notin W^\perp \iff \Rad (\restr{\varphi}{W}) = W \cap W^\perp \iff \vec w \text{ non è isotropo } = \zerovecset \iff - V = W \oplus W^\perp$. -\end{remark} - -\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 per cui la matrice associata del prodotto scalare è diagonale. -\end{definition} - -\begin{proposition} (formula di polarizzazione) - Se $\Char \KK \neq 2$, un prodotto scalare è univocamente determinato dalla sua forma quadratica $q$. -\end{proposition} - -\begin{proof} - Si nota infatti che $q(\vec v + \vec w) - q(\vec v) - q(\vec w) = 2 \varphi(\vec v, \vec w)$, e quindi, - poiché $2$ è invertibile per ipotesi, che $\varphi(\vec v, \vec w) = 2\inv (q(\vec v + \vec w) - q(\vec v) - q(\vec w))$. -\end{proof} - -\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} - Sia dimostra il teorema per induzione su $n := \dim V$. Per $n \leq 1$, la dimostrazione è triviale. 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. -\end{proof} - - -\begin{note} - D'ora in poi, nel corso del documento, si assumerà $\Char \KK \neq 2$. -\end{note} - -\begin{theorem} (di Sylvester, caso complesso) - Sia $\KK$ un campo i cui elementi sono tutti quadrati di un - altro elemento del campo (e.g.~$\CC$). Allora esiste una base - ortogonale $\basis$ tale per cui: - - \[ M_\basis(\varphi) = \Matrix{I_r & \rvline & 0 \\ \hline 0 & \rvline & 0\,}. \] -\end{theorem} - -\begin{proof} - Per il teorema di Lagrange, esiste una base ortogonale $\basis'$ di $V$. - Si riordini la base in modo tale che la forma quadratica valutata nei primi elementi sia sempre diversa da zero. Allora, poiché ogni - elemento di $\KK$ è per ipotesi quadrato di un altro elemento - di $\KK$, si sostituisca $\basis'$ con una base $\basis$ tale per - cui, se $q(\vv i) = 0$, $\vv i \mapsto \vv i$, e altrimenti - $\vv i \mapsto \frac{\vv i}{\sqrt{q(\vv i)}}$. Allora $\basis'$ - è una base tale per cui la matrice associata del prodotto scalare - in tale base è proprio come desiderata nella tesi, dove $r$ è - il numero di elementi tali per cui la forma quadratica valutata - in essi sia diversa da zero. -\end{proof} - -\begin{remark} - Si possono effettuare alcune considerazioni sul teorema di Sylvester - complesso. \\ - - \li Si può immediatamente concludere che il rango è un invariante - completo per la congruenza in un campo in cui tutti gli elementi - sono quadrati, ossia che $A \cong B \iff \rg(A) = \rg(B)$, se $A$ e - $B$ sono matrici simmetriche: infatti - ogni matrice simmetrica rappresenta una prodotto scalare, ed è - pertanto congruente ad una matrice della forma desiderata - nell'enunciato del teorema di Sylvester complesso. Poiché il rango - è un invariante della congruenza, si ricava che $r$ nella forma - della matrice di Sylvester, rappresentando il rango, è anche - il rango di ogni sua matrice congruente. In particolare, se due - matrici simmetriche hanno stesso rango, allora sono congruenti - alla stessa matrice di Sylvester, e quindi, essendo la congruenza - una relazione di congruenza, sono congruenti a loro volta. \\ - \li Due matrici simmetriche con stesso rango, allora, non solo - sono SD-equivalenti, ma sono anche congruenti. \\ - \li Ogni base ortogonale deve quindi avere lo stesso numero - di elementi nulli. -\end{remark} - -\begin{definition} (somma diretta ortogonale) - Siano i sottospazi $U$ e $W \subseteq V$ in somma diretta. Allora si dice che $U$ e $W$ sono in \textbf{somma - diretta ortogonale rispetto al prodotto scalare} $\varphi$ di $V$, ossia che $U \oplus W = U \oplus^\perp W$, se $\varphi(\vec u, \vec w) = 0$ $\forall \vec u \in U$, $\vec w \in W$. -\end{definition} - -\begin{definition} (cono isotropo) - Si definisce \textbf{cono isotropo} di $V$ rispetto al prodotto scalare $\varphi$ il seguente insieme: - - \[ \CI(\varphi) = \{ \v \in V \mid \varphi(\v, \v) = 0 \}, \] - - \vskip 0.05in - - ossia l'insieme dei vettori isotropi di $V$. -\end{definition} - -\begin{note} - La notazione $\varphi > 0$ indica che $\varphi$ è definito positivo (si scrive $\varphi \geq 0$ se invece è semidefinito - positivo). - Analogamente $\varphi < 0$ indica che $\varphi$ è definito negativo (e $\varphi \leq 0$ indica che è semidefinito negativo). -\end{note} - -\begin{exercise} Sia $\Char \KK \neq 2$. - Siano $\vv1$, ..., $\vv k \in V$ e sia $M = \left( \varphi(\vv i, \vv j) \right)_{i, j = 1\textrm{---}k} \in M(k, \KK)$, - dove $\varphi$ è un prodotto scalare di $V$. Sia inoltre $W = \Span(\vv 1, ..., \vv k)$. Si dimostrino - allora le seguenti affermazioni. - - \begin{enumerate}[(i)] - \item Se $M$ è invertibile, allora $\vv 1$, ..., $\vv k$ sono linearmente indipendenti. - - \item Siano $\vv 1$, ..., $\vv k$ linearmente indipendenti. Allora $M$ è invertibile $\iff$ $\restr{\varphi}{W}$ è non degenere $\iff$ $W \cap W^\perp = \zerovecset$. - - \item Siano $\vv1$, ..., $\vv k$ a due a due ortogonali tra loro. Allora $M$ è invertibile $\iff$ nessun - vettore $\vv i$ è isotropo. - - \item Siano $\vv1$, ..., $\vv k$ a due a due ortogonali tra loro e siano anche linearmente indipendenti. - Allora $M$ è invertibile $\implies$ si può estendere $\basis_W = \{\vv 1, \ldots, \vv k\}$ a una base ortogonale di $V$. - - \item Sia $\KK = \RR$. Sia inoltre $\varphi > 0$. Allora $\vv 1$, ..., $\vv k$ sono linearmente - indipendenti $\iff$ $M$ è invertibile. - - \item Sia $\KK = \RR$. Sia ancora $\varphi > 0$. Allora se $\vv 1$, ..., $\vv k$ sono a due a due - ortogonali e sono tutti non nulli, sono anche linearmente indipendenti. - \end{enumerate} -\end{exercise} - -\begin{solution} - \begin{enumerate}[(i)] - \item Siano $a_1$, ..., $a_k \in \KK$ tali che $a_1 \vv 1 + \ldots + a_k \vv k = 0$. Vale in - particolare che $\vec 0 = \varphi(\vv i, \vec 0) = \varphi(\vv i, a_1 \vv 1 + \ldots + a_k \vv k) = - \sum_{j=1}^k a_j \varphi(\vv i, \vv j)$ $\forall 1 \leq i \leq k$. Allora $\sum_{j=1}^k a_j M^j = 0$. - Dal momento che $M$ è invertibile, $\rg(M) = k$, e quindi l'insieme delle colonne di $M$ è linearmente - indipendente, da cui si ricava che $a_j = 0$ $\forall 1 \leq j \leq k$, e quindi che $\vv 1$, ..., - $\vv k$ sono linearmente indipendenti. - - \item Poiché $\vv 1$, ..., $\vv k$ sono linearmente indipendenti, tali vettori formano una base di - $W$, detta $\basis$. In particolare, allora, vale che $M = M_\basis(\restr{\varphi}{W})$. Pertanto, - se $M$ è invertibile, $\Rad(\restr{\varphi}{W}) = \Ker M = \zerovecset$, e dunque $\restr{\varphi}{W}$ - è non degenere. Se invece $\restr{\varphi}{W}$ è non degenere, $\zerovecset = \Rad(\restr{\varphi}{W}) = W \cap W^\perp$. Infine, se $W \cap W^\perp = \zerovecset$, $\zerovecset = W \cap W^\perp = \Rad(\restr{\varphi}{W}) = \Ker M$, e quindi $M$ è iniettiva, e dunque invertibile. - - \item Dal momento che $\vv 1$, ..., $\vv k$ sono ortogonali tra loro, $M$ è una matrice diagonale. - Pertanto $M$ è invertibile se e solo se ogni suo elemento diagonale è diverso da $0$, ossia - se $\varphi(\vv i, \vv i) \neq 0$ $\forall 1 \leq i \leq k$, e dunque se e solo se nessun vettore - $\vv i$ è isotropo. - - \item Se $M$ è invertibile, da (ii) si deduce che $\Rad(\restr{\varphi}{W}) = W \cap W^\perp = \zerovecset$, - e quindi che $W$ e $W^\perp$ sono in somma diretta. Inoltre, per la formula delle dimensioni del prodotto - scalare, $\dim W + \dim W^\perp = \dim V + \underbrace{\dim (W \cap V^\perp)}_{\leq \dim (W \cap W^\perp) = 0} = \dim V$. Pertanto $V = W \oplus^\perp W^\perp$. \\ - - Allora, dacché $\Char \KK \neq 2$, per il teorema di Lagrange, $W^\perp$ ammette una base ortogonale $\basis_{W^\perp}$. Si conclude - dunque che $\basis = \basis_W \cup \basis_{W^\perp}$ è una base ortogonale di $V$. - - \item Se $M$ è invertibile, da (i) $\vv1$, ..., $\vv k$ sono linearmente indipendenti. Siano ora - invece $\vv 1$, ..., $\vv k$ linearmente indipendenti per ipotesi. Siano $a_1$, ..., $a_k \in \KK$ tali - che $a_1 M^1 + \ldots + a_k M^k = 0$, allora $a_1 \varphi(\vv i, \vv 1) + \ldots + a_k \varphi(\vv i, \vv k) = 0$ $\forall 1 \leq i \leq k$. Pertanto, detto $\v = a_1 \vv 1 + \ldots + a_k \vv k$, si ricava che: - - \[ \varphi(\v, \v) = \sum_{i=1}^k \sum_{j=1}^k a_j \, \varphi(\vv i, \vv j) = 0. \] - - Tuttavia questo è possibile solo se $\v = a_1 \vv 1 + \ldots + a_k \vv k = 0$. Dal momento che - $\vv 1$, ..., $\vv k$ sono linearmente indipendenti, si conclude che $a_1 = \cdots = a_k = 0$, ossia - che le colonne di $M$ sono tutte linearmente indipendenti e quindi che $\rg(M) = k \implies$ $M$ è invertibile. - - \item Poiché $\vv 1$, ..., $\vv k$ sono ortogonali a due a due tra loro, $M$ è una matrice diagonale. - Inoltre, dacché $\varphi > 0$ e $\vv i \neq \vec 0$ $\forall 1 \leq i \leq k$, gli elementi diagonali di $M$ sono sicuramente tutti diversi da zero, e quindi $\det (M) \neq 0$ $\implies$ $M$ è invertibile. Allora, - per il punto (v), $\vv 1$, ..., $\vv k$ sono linearmente indipendenti. - \end{enumerate} -\end{solution} - -\begin{definition} - Data una base ortogonale $\basis$ di $V$ rispetto al prodotto - scalare $\varphi$, - si definiscono i seguenti indici: - - \begin{align*} - \iota_+(\varphi) &= \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} > 0 \}, &\text{(}\textbf{indice di positività}\text{)} \\ - \iota_-(\varphi) &= \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} < 0 \}, &\text{(}\textbf{indice di negatività}\text{)}\\ - \iota_0(\varphi) &= \dim V^\perp &\text{(}\textbf{indice di nullità}\text{)} - \end{align*} - - Quando il prodotto scalare $\varphi$ è noto dal contesto, si omette - e si scrive solo $\iota_+$, $\iota_-$ e $\iota_0$. In particolare, - la terna $\sigma(\varphi) = \sigma = (i_+, i_-, i_0)$ è detta \textbf{segnatura} del - prodotto $\varphi$. -\end{definition} - -\begin{theorem} (di Sylvester, caso reale) Sia $\KK$ un campo ordinato - i cui elementi positivi sono tutti quadrati (e.g.~$\RR$). Allora - esiste una base ortogonale $\basis$ tale per cui: - - \[ M_\basis(\varphi) = \Matrix{I_{\iota_+} & \rvline & 0 & \rvline & 0 \\ \hline 0 & \rvline & -I_{\iota_-} & \rvline & 0 \\ \hline 0 & \rvline & 0 & \rvline & 0\cdot I_{\iota_0} }. \] - - \vskip 0.05in - - Inoltre, per ogni base ortogonale, esistono esattamente - $\iota_+$ vettori della base con forma quadratica positiva, - $\iota_-$ con forma negativa e $\iota_0$ con - forma nulla. -\end{theorem} - -\begin{proof} - Per il teorema di Lagrange, esiste una base ortogonale $\basis'$ di $V$. - Si riordini la base in modo tale che la forma quadratica valutata nei primi elementi sia strettamente positiva, che nei secondi elementi sia strettamente negativa e che negli ultimi sia nulla. Si sostituisca - $\basis'$ con una base $\basis$ tale per cui, se $q(\vv i) > 0$, - allora $\vv i \mapsto \frac{\vv i}{\sqrt{q(\vv i)}}$; se - $q(\vv i) < 0$, allora $\vv i \mapsto \frac{\vv i}{\sqrt{-q(\vv i)}}$; - altrimenti $\vv i \mapsto \vv i$. Si è allora trovata una base - la cui matrice associata del prodotto scalare è come desiderata nella - tesi. \\ - - Sia ora $a$ il numero di vettori della base con forma quadratica - positiva, $b$ il numero di vettori con forma negativa e $c$ quello - dei vettori con forma nulla. Si consideri $W_+ = \Span(\vv 1, ..., \vv a)$, $W_- = \Span(\vv{a+1}, ..., \vv b)$, $W_0 = \Span(\vv{b+1}, ..., \vv c)$. \\ - - Sia $M = M_\basis(\varphi)$. Si osserva che $c = n - \rg(M) = \dim \Ker(M) = \dim V^\perp = \iota_0$. Inoltre $\forall \v \in W_+$, dacché - $\basis$ è ortogonale, - $q(\v) = q(\sum_{i=1}^a \alpha_i \vv i) = \sum_{i=1}^a \alpha_i^2 q(\vv i) > 0$, e quindi $\restr{\varphi}{W_+} > 0$, da cui $\iota_+ \geq a$. - Analogamente $\iota_- \geq b$. \\ - - Si mostra ora che è impossibile che $\iota_+ > a$. Se così infatti - fosse, sia $W$ tale che $\dim W = \iota_+$ e che $\restr{\varphi}{W} > 0$. $\iota_+ + b + c$ sarebbe maggiore di $a + b + c = n := \dim V$. Quindi, per la formula di Grassman, $\dim(W + W_- + W_0) = \dim W + - \dim(W_- + W_0) - \dim (W \cap (W_- + W_0)) \implies \dim (W \cap (W_- + W_0)) = \dim W + - \dim(W_- + W_0) - \dim(W + W_- + W_0) > 0$, ossia esisterebbe - $\v \neq \{\vec 0\} \mid \v \in W \cap (W_- + W_0)$. Tuttavia - questo è assurdo, dacché dovrebbe valere sia $q(\v) > 0$ che - $q(\v) < 0$, \Lightning. Quindi $\iota_+ = a$, e analogamente - $\iota_- = b$. -\end{proof} - -\begin{definition} - Si dice \textbf{base di Sylvester} una base di $V$ tale per cui la - matrice associata di $\varphi$ sia esattamente nella forma - vista nella dimostrazione del teorema di Sylvester. Analogamente - si definisce tale matrice come \textbf{matrice di Sylvester}. -\end{definition} - -\begin{remark} \nl - \li Si può dunque definire la segnatura di una matrice simmetrica - come la segnatura di una qualsiasi sua base ortogonale, dal - momento che tale segnatura è invariante per cambiamento di base. \\ - \li La segnatura è un invariante completo per la congruenza nel caso reale. Se infatti due matrici hanno la stessa segnatura, sono - entrambe congruenti alla matrice come vista nella dimostrazione - della forma reale del teorema di Sylvester, e quindi, essendo - la congruenza una relazione di equivalenza, sono congruenti - tra loro. Analogamente vale il viceversa, dal momento che ogni - base ortogonale di due matrici congruenti devono contenere gli - stessi numeri $\iota_+$, $\iota_-$ e $\iota_0$ di vettori - di base con forma quadratica positiva, negativa e nulla. \\ - \li Se $\ww 1$, ..., $\ww k$ sono tutti i vettori di una base - ortogonale $\basis$ con forma quadratica nulla, si osserva che $W = \Span(\ww 1, ..., \ww k)$ altro non è che $V^\perp$ stesso. Infatti, come - visto anche nella dimostrazione del teorema di Sylvester reale, vale - che $\dim W = \dim \Ker (M_\basis(\varphi)) = \dim V^\perp$. Inoltre, - se $\w \in W$ e $\v \in V$, $\varphi(\w, \v) = \varphi(\sum_{i=1}^k - \alpha_i \ww i, \sum_{i=1}^k \beta_i \ww i + \sum_{i=k+1}^n \beta_i \vv i) - = \sum_{i=1}^k \alpha_i \beta_i q(\ww i) = 0$, e quindi - $W \subseteq V^\perp$, da cui si conclude che $W = V^\perp$. \\ - \li Vale in particolare che $\rg(\varphi) = \iota_+ + \iota_-$, mentre - $\dim \Ker(\varphi) = \iota_0$, e quindi $n = \iota_+ + \iota_- + \iota_0$. \\ - \li Se $V = U \oplusperp W$, allora $\iota_+(\varphi) = \iota_+(\restr{\varphi}{V}) + \iota_+(\restr{\varphi}{W})$, e - analogamente per gli altri indici. -\end{remark} - -\begin{definition} (isometria) - Dati due spazi vettoriali $(V, \varphi)$ e - $(V', \varphi')$ dotati di prodotto scalare sullo stesso campo $\KK$, si dice che - $V$ e $V'$ sono \textbf{isometrici} se esiste un isomorfismo - $f$, detto \textit{isometria}, che preserva tali che prodotti, ossia tale che: - - \[ \varphi(\vec v, \vec w) = \varphi'(f(\vec v), f(\vec w)). \] -\end{definition} - -\begin{exercise} Sia $f : V \to V'$ un isomorfismo. Allora $f$ è un'isometria $\iff$ $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$ $\iff$ $\exists$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$ tale che $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. -\end{exercise} - -\begin{solution} Se $f$ è un'isometria, detta $\basis$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$ - dal momento che $f$ è anche un isomorfismo. Inoltre, dacché $f$ è un'isometria, vale sicuramente che - $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. \\ - - Sia ora assunto per ipotesi che $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Allora, analogamente a prima, detta $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$, e in quanto tale, - per ipotesi, è tale che $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. \\ - - Sia infine assunto per ipotesi che $\exists$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$ tale che $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Siano $\v$, $\w \in V$. Allora $\exists a_1$, ..., $a_n$, $b_1$, ..., $b_n \in \KK$ - tali che $\v = a_1 \vv 1 + \ldots + a_n \vv n$ e $\w = b_1 \vv 1 + \ldots + b_n \vv n$. Si ricava pertanto - che: - - \[ \varphi'(f(\v), f(\w)) = \sum_{i=1}^n \sum_{j=1}^n a_i b_j \, \varphi'(f(\vv i), f(\vv j)) = - \sum_{i=1}^n \sum_{j=1}^n a_i b_j \, \varphi(\vv i, \vv j) = \varphi(\v, \w), \] - - da cui la tesi. -\end{solution} - -\begin{proposition} Sono equivalenti le seguenti affermazioni: - - \begin{enumerate}[(i)] - \item $V$ e $V'$ sono isometrici; - \item $\forall$ base $\basis$ di $V$, base $\basis'$ di $V'$, - $M_\basis(\varphi)$ e $M_{\basis'}(\varphi')$ sono congruenti; - \item $\exists$ base $\basis$ di $V$, base $\basis'$ di $V'$ tale che - $M_\basis(\varphi)$ e $M_{\basis'}(\varphi')$ sono congruenti. - \end{enumerate} -\end{proposition} - -\begin{proof} Se $V$ e $V'$ sono isometrici, sia $f : V \to V'$ un'isometria. Sia $\basisC = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Allora, poiché $f$ è anche un isomorfismo, $\basisC' = f(\basisC)$ è una base di $V$ tale che - $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Pertanto $M_\basisC(\varphi) = M_{\basisC'}(\varphi')$. Si conclude allora che, cambiando base in $V$ (o in $V'$), la matrice associata - al prodotto scalare varia per congruenza dalla formula di cambiamento di base per il prodotto scalare, da cui si ricava che per ogni scelta di $\basis$ base di $V$ e di $\basis'$ base di $V'$, $M_\basis(\varphi) \cong M_{\basis'}(\varphi')$. Inoltre, se tale risultato è vero per ogni $\basis$ base di $V$ e di $\basis'$ base di $V'$, dal momento che sicuramente esistono due basi $\basis$, $\basis'$ di $V$ e $V'$, vale anche (ii) $\implies$ (iii). \\ - - Si dimostra ora (iii) $\implies$ (i). Per ipotesi $M_\basis(\varphi) \cong M_{\basis'}(\varphi')$, quindi - $\exists P \in \GL(n, \KK) \mid M_{\basis'}(\varphi') = P^\top M_\basis(\varphi) P$. Allora $\exists$ $\basis''$ - base di $V'$ tale che $P = M_{\basis''}^{\basis'}(\Idv)$, da cui $P\inv = M_{\basis'}^{\basis''}(\varphi)$. Per la formula di cambiamento di base del prodotto - scalare, $M_{\basis''}(\varphi) = (P\inv)^\top M_{\basis'} P\inv = M_\basis(\varphi)$. Detta - $\basis'' = \{ \ww 1, \ldots, \ww n \}$, si costruisce allora l'isomorfismo $f : V \to V'$ tale - che $f(\vv i) = \ww i$ $\forall 1 \leq i \leq n$.. Dal momento che per costruzione $M_\basis(\varphi) = M_{\basis''}(\varphi')$, - $\varphi(\vv i, \vv j) = \varphi'(\ww i, \ww j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. - Si conclude dunque che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v, \w \in V$, e dunque - che $f$ è un'isometria, come desiderato dalla tesi. -\end{proof} - -\begin{proposition} $(V, \varphi)$ e $(V', \varphi')$ spazi vettoriali - su $\RR$ sono - isometrici $\iff$ $\varphi$ e $\varphi'$ hanno la stessa segnatura. -\end{proposition} - -\begin{proof}\nl\nl - \rightproof Per la precedente proposizione, esistono due basi $\basis$ e $\basis'$, una di $V$ e una di $V'$, - tali che $M_\basis(\varphi) \cong M_{\basis'}(\varphi)$. Allora queste due matrici condividono la stessa - segnatura, e così quindi anche $\varphi$ e $\varphi'$. \\ - - \leftproof Se $\varphi$ e $\varphi'$ hanno la stessa segnatura, esistono due basi $\basis = \{ \vv 1, \ldots, \vv n \}$ e $\basis' = \{ \ww 1, \ldots, \ww n \}$, una - di $V$ e una di $V'$, tali che $M = M_\basis(\varphi) = M_{\basis'}(\varphi')$ e che $M$ è una matrice di - Sylvester. Allora si costruisce $f : V \to V'$ tale che $f(\vv i) = \ww i$. Esso è un isomorfismo, e per - costruzione $\varphi(\vv i, \vv j) = \varphi'(\ww i, \ww j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$, da cui - si conclude che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v$, $\w \in V$, e quindi che $V$ e - $V'$ sono isometrici. -\end{proof} - -% \begin{example} - % Per $\varphi = x_1 y_1 + x_2 y_2 - x_3 y_3$. %TODO: guarda dalle slide. - % \end{example} - -\begin{definition} (sottospazio isotropo) - Sia $W$ un sottospazio di $V$. Allora $W$ si dice \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} - 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 $V$, $W \subseteq W^\perp \implies \dim W \leq \dim W^\perp$. - Allora, poiché $\varphi$ è non degenere, $\dim W + \dim W^\perp = \dim V$, $\dim W \leq \dim V - \dim W$, - da cui $\dim W \leq \frac{1}{2} \dim V$. -\end{proof} - -\begin{definition} - Si definisce \textbf{indice di Witt} $W(\varphi)$ di $(V, \varphi)$ - come la massima dimensione di un sottospazio isotropo. -\end{definition} - -\begin{remark}\nl - \li Se $\varphi > 0$ o $\varphi < 0$, $W(\varphi) = 0$. -\end{remark} - -\begin{proposition} - Sia $\KK = \RR$. Sia $\varphi$ non degenere e sia $\sigma(\varphi) = (\iota_+(\varphi), \iota_-(\varphi), 0)$. 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, $\dim W + \dim W^+ > n \implies \dim W + \dim W^+ > \dim W + \dim W^+ - \dim (W \cap W^+) \implies \dim (W \cap W^+) > 0$. 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 tale per cui $M_\basis(\varphi)$ è la matrice 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 $W(\varphi) = i_-(\varphi)$, da cui la tesi. -\end{proof} \ No newline at end of file diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/3. Prodotti hermitiani, spazi euclidei e teorema spettrale.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/3. Spazi euclidei e teorema spettrale.tex similarity index 80% rename from Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/3. Prodotti hermitiani, spazi euclidei e teorema spettrale.tex rename to Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/3. Spazi euclidei e teorema spettrale.tex index 8b074c0..8677e39 100644 --- a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/3. Prodotti hermitiani, spazi euclidei e teorema spettrale.tex +++ b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/3. Spazi euclidei e teorema spettrale.tex @@ -1,4 +1,4 @@ -\chapter{Prodotti hermitiani, spazi euclidei e teorema spettrale (non indicizzato)} +\chapter{Spazi euclidei e teorema spettrale (non indicizzato)} \begin{note} Nel corso del documento, per $V$ si intenderà uno spazio vettoriale di dimensione @@ -6,327 +6,6 @@ dipendentemente dal contesto. \end{note} -\begin{definition} (prodotto hermitiano) Sia $\KK = \CC$. Una mappa $\varphi : V \times V \to \CC$ si dice \textbf{prodotto hermitiano} se: - - \begin{enumerate}[(i)] - \item $\varphi$ è $\CC$-lineare nel secondo argomento, ossia se $\varphi(\v, \U + \w) = \varphi(\v, \U) + \varphi(\v, \w)$ e - $\varphi(\v, a \w) = a \, \varphi(\v, \w)$, - \item $\varphi(\U, \w) = \conj{\varphi(\w, \U)}$. - \end{enumerate} -\end{definition} - -\begin{definition} (prodotto hermitiano canonico in $\CC^n$) Si definisce - \textbf{prodotto hermitiano canonico} di $\CC^n$ il prodotto $\varphi : \CC^n \times \CC^n \to \CC$ tale per cui, detti $\v = (z_1 \cdots z_n)^\top$ e $\w = (w_1 \cdots w_n)^\top$, $\varphi(\v, \w) = \sum_{i=1}^n \conj{z_i} w_i$. -\end{definition} - -\begin{remark}\nl - \li $\varphi(\U + \w, \v) = \conj{\varphi(\v, \U + \w)} = - \conj{\varphi(\v, \U) + \varphi(\v, \w)} = \conj{\varphi(\v, \U)} + \conj{\varphi(\v, \U)} = \varphi(\w, \v) + \varphi(\U, \v)$, ossia - $\varphi$ è additiva anche nel primo argomento. \\ - \li $\varphi(a \v, \w) = \conj{\varphi(\w, a \v)} = \conj{a} \conj{\varphi(\w, \v)} = \conj{a} \, \varphi(\v, \w)$. \\ - \li $\varphi(\v, \v) = \conj{\varphi(\v, \v)}$, e quindi $\varphi(\v, \v) \in \RR$. \\ - \li Sia $\v = \sum_{i=1}^n x_i \vv i$ e sia $\w = \sum_{i=1}^n y_i \vv i$, allora $\varphi(\v, \w) = \sum_{i =1}^n \sum_{j=1}^n \conj{x_i} y_i \varphi(\vv i, \vv j)$. \\ - \li $\varphi(\v, \w) = 0 \iff \varphi(\w, \v) = 0$. -\end{remark} - -\begin{proposition} - Data la forma quadratica $q : V \to \RR$ del prodotto hermitiano $\varphi$ tale che $q(\v) = \varphi(\v, \v) \in \RR$, tale - forma quadratica individua univocamente il prodotto hermitiano $\varphi$. -\end{proposition} - -\begin{proof} - Innanzitutto si osserva che: - - \[ \varphi(\v, \w) = \frac{\varphi(\v, \w) + \conj{\varphi(\v, \w)}}{2} + \frac{\varphi(\v, \w) . \conj{\varphi(\v, \w)}}{2}. \] - - \vskip 0.05in - - Si considerano allora le due identità: - - \[ q(\v + \w) - q(\v) - q(\w) = - \varphi(\v, \w) + \conj{\varphi(\w, \v)} = 2 \, \Re(\varphi(\v, \w)), \] - - \[ q(i\v + \w) - q(\v) - q(\w) = -i(\varphi(\v, \w) - \conj{\varphi(\v, \w)}) = 2 \, \imm(\varphi(\v, \w)), \] - - \vskip 0.05in - - da cui si conclude che il prodotto $\varphi$ è univocamente - determinato dalla sua forma quadratica. -\end{proof} - -\begin{definition} - Si definisce \textbf{matrice aggiunta} di $A \in M(n, \KK)$ la matrice coniugata della trasposta di $A$, ossia: - - \[ A^* = \conj{A^\top} = \conj{A}^\top. \] -\end{definition} - -\begin{remark} - Per quanto riguarda la matrice aggiunta valgono le principali proprietà della matrice trasposta: - - \begin{itemize} - \item $(A + B)^* = A^* + B^*$, - \item $(AB)^* = B^* A^*$, - \item $(A\inv)^* = (A^*)\inv$, se $A$ è invertibile. - \end{itemize} -\end{remark} - -%TODO: aggiungere tr(conj(A^t) B) - -\begin{definition} (matrice associata del prodotto hermitiano) Analogamente - al caso del prodotto scalare, data una base $\basis = \{\vv 1, \ldots, \vv n\}$ si definisce - come \textbf{matrice associata del prodotto hermitiano} $\varphi$ - la matrice $M_\basis(\varphi) = (\varphi(\vv i, \vv j))_{i,j = 1 \textrm{---} n}$. -\end{definition} - -\begin{remark} - Si osserva che, analogamente al caso del prodotto scalare, vale - la seguente identità: - - \[ \varphi(\v, \w) = [\v]_\basis^* M_\basis(\varphi) [\w]_\basis. \] -\end{remark} - -\begin{proposition} - (formula del cambiamento di base per i prodotto hermitiani) Siano - $\basis$, $\basis'$ due basi di $V$. Allora vale la seguente - identità: - - \[ M_{\basis'} = M_{\basis}^{\basis'}(\Idv)^* M_\basis(\varphi) M_{\basis}^{\basis'}(\Idv). \] -\end{proposition} - -\begin{proof} - Siano $\basis = \{ \vv 1, \ldots, \vv n \}$ e $\basis' = \{ \ww 1, \ldots, \ww n \}$. Allora $\varphi(\ww i, \ww j) = [\ww i]_\basis^* M_\basis(\varphi) [\ww j]_\basis = \left( M_\basis^{\basis'}(\Idv)^i \right)^* M_\basis(\varphi) M_\basis^{\basis'}(\Idv)^j = - \left(M_\basis^{\basis'}(\Idv)\right)^*_i M_\basis(\varphi) M_\basis^{\basis'}(\Idv)^j$, da cui si ricava l'identità - desiderata. -\end{proof} - -\begin{definition} (radicale di un prodotto hermitiano) - Analogamente al caso del prodotto scalare, si definisce il \textbf{radicale} del prodotto $\varphi$ come il seguente sottospazio: - - \[ V^\perp = \{ \v \in V \mid \varphi(\v, \w) = 0 \, \forall \w \in V \}. \] -\end{definition} - -\begin{proposition} - Sia $\basis$ una base di $V$ e $\varphi$ un prodotto hermitiano. Allora $V^\perp = [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$\footnote{Stavolta non è sufficiente considerare la mappa $f : V \to V^*$ tale che $f(\v) = \left[ \w \mapsto \varphi(\v, \w) \right]$, dal momento che $f$ non è lineare, bensì antilineare, ossia $f(a \v) = \conj a f(\v)$.}. -\end{proposition} - -\begin{proof} - Sia $\basis = \{ \vv 1, \ldots, \vv n \}$ e sia $\v \in V^\perp$. - Siano $a_1$, ..., $a_n \in \KK$ tali che $\v = a_1 \vv 1 + \ldots + a_n \vv n$. Allora, poiché $\v \in V$, $0 = \varphi(\vv i, \v) - = a_1 \varphi(\vv i, \vv 1) + \ldots + a_n \varphi(\vv i, \vv n) = M_i [\v]_\basis$, da cui si ricava che $[\v]_\basis \in \Ker M_\basis(\varphi)$, e quindi che $V^\perp \subseteq [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$. \\ - - Sia ora $\v \in V$ tale che $[\v]_\basis \in \Ker M_\basis(\varphi)$. - Allora, per ogni $\w \in V$, $\varphi(\w, \v) = [\w]_\basis^* M_\basis(\varphi) [\v]_\basis = [\w]_\basis^* 0 = 0$, da cui si - conclude che $\v \in V^\perp$, e quindi che $V^\perp \supseteq [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$, da cui - $V^\perp = [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$, ossia - la tesi. -\end{proof} - -\begin{remark} - Come conseguenza della proposizione appena dimostrata, valgono - le principali proprietà già viste per il prodotto scalare. \\ - - \li $\det(M_\basis(\varphi)) = 0 \iff V^\perp \neq \zerovecset \iff \varphi$ è degenere, \\ - \li Vale il teorema di Lagrange, e quindi quello di Sylvester, benché con alcune accortezze: si - introduce, come nel caso di $\RR$, il concetto di segnatura, che diventa l'invariante completo - della nuova congruenza hermitiana, che ancora una volta si dimostra essere una relazione - di equivalenza. \\ - \li Come mostrato nei momenti finali del documento (vd.~\textit{Esercizio 3}), vale - la formula delle dimensioni anche nel caso del prodotto hermitiano. -\end{remark} - -\hr - -\begin{definition} (restrizione ai reali di uno spazio) Sia $V$ - uno spazio vettoriale su $\CC$ con base $\basis$. Si definisce allora lo spazio $V_\RR$, detto - \textbf{spazio di restrizione su $\RR$} di $V$, come uno spazio su $\RR$ generato da - $\basis_\RR = \basis \cup i \basis$. -\end{definition} - -\begin{example} - Si consideri $V = \CC^3$. Una base di $\CC^3$ è chiaramente $\{ \e1, \e2, \e3 \}$. Allora - $V_\RR$ sarà uno spazio vettoriale su $\RR$ generato dai vettori $\{ \e1, \e2, \e3, i\e1, i\e2, i\e3 \}$. -\end{example} - -\begin{remark} - Si osserva che lo spazio di restrizione su $\RR$ e lo spazio di partenza condividono lo stesso insieme - di vettori. Infatti, $\Span_\CC(\basis) = \Span_\RR(\basis \cup i\basis)$. Ciononostante, $\dim V_\RR = 2 \dim V$\footnote{Si sarebbe potuto ottenere lo stesso risultato utilizzando il teorema delle torri algebriche: $[V_\RR : \RR] = [V: \CC] [\CC: \RR] = 2 [V : \CC]$.}, se $\dim V \in \NN$. -\end{remark} - -\begin{definition} (complessificazione di uno spazio) Sia $V$ uno spazio vettoriale su $\RR$. - Si definisce allora lo \textbf{spazio complessificato} $V_\CC = V \times V$ su $\CC$ con le seguenti operazioni: - - \begin{itemize} - \item $(\v, \w) + (\v', \w') = (\v + \v', \w + \w')$, - \item $(a+bi)(\v, \w) = (a\v - b\w, a\w + b\v)$. - \end{itemize} -\end{definition} - -\begin{remark} - La costruzione dello spazio complessificato emula in realtà la costruzione di $\CC$ come spazio - $\RR \times \RR$. Infatti se $z = (c, d)$, vale che $(a + bi)(c, d) = (ac - bd, ad + bc)$, mentre - si mantiene l'usuale operazione di addizione. In particolare si può identificare l'insieme - $V \times \zerovecset$ come $V$, mentre $\zerovecset \times V$ viene identificato come l'insieme - degli immaginari $iV$ di $V_\CC$. Infine, moltiplicare per uno scalare reale un elemento di - $V \times \zerovecset$ equivale a moltiplicare la sola prima componente con l'usuale operazione - di moltiplicazione di $V$. Allora, come accade per $\CC$, si può sostituire la notazione - $(\v, \w)$ con la più comoda notazione $\v + i \w$. -\end{remark} - -\begin{remark} - Sia $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Innanzitutto si osserva che - $(a+bi)(\v, \vec 0) = (a\v, b\v)$. Pertanto si può concludere che $\basis \times \zerovecset$ è - una base dello spazio complessificato $V_\CC$ su $\CC$. \\ - - Infatti, se $(a_1 + b_1 i)(\vv 1, \vec 0) + \ldots + (a_n + b_n i)(\vv n, \vec 0) = (\vec 0, \vec 0)$, - allora $(a_1 \vv 1 + \ldots + a_n \vv n, b_1 \vv 1 + \ldots + b_n \vv n) = (\vec 0, \vec 0)$. - Poiché però $\basis$ è linearmente indipendente per ipotesi, l'ultima identità implica che - $a_1 = \cdots = a_n = b_1 = \cdots = b_n = 0$, e quindi che $\basis \times \zerovecset$ è linearmente - indipendente. \\ - - Inoltre $\basis \times \zerovecset$ genera $V_\CC$. Se infatti $\v = (\U, \w)$, e vale che: - - \[ \U = a_1 \vv 1 + \ldots + a_n \vv n, \quad \w = b_1 \vv 1 + \ldots + b_n \vv n, \] - - \vskip 0.1in - - allora $\v = (a_1 + b_1 i) (\vv 1, \vec 0) + \ldots + (a_n + b_n i) (\vv n, \vec 0)$. Quindi - $\dim V_\CC = \dim V$. -\end{remark} - -\begin{definition} - Sia $f$ un'applicazione $\CC$-lineare di $V$ spazio vettoriale su $\CC$. Allora - si definisce la \textbf{restrizione su} $\RR$ di $f$, detta $f_\RR : V_\RR \to V_\RR$, - in modo tale che $f_\RR(\v) = f(\v)$. -\end{definition} - -\begin{remark} - Sia $\basis = \{\vv 1, \ldots, \vv n\}$ una base di $V$ su $\CC$. Sia $A = M_\basis(f)$. Si - osserva allora che, se $\basis' = \basis \cup i \basis$ e $A = A' + i A''$ con $A'$, $A'' \in M(n, \RR)$, - vale la seguente identità: - - \[ M_{\basis'}(f_\RR) = \Matrix{ A' & \rvline & -A'' \\ \hline A'' & \rvline & A' }. \] - - Infatti, se $f(\vv i) = (a_1 + b_1 i) \vv 1 + \ldots + (a_n + b_n i) \vv n$, vale che - $f_\RR(\vv i) = a_1 \vv 1 + \ldots + a_n \vv n + b_1 (i \vv 1) + \ldots + b_n (i \vv n)$, - mentre $f_\RR(i \vv i) = i f(\vv i) = - b_1 \vv 1 + \ldots - b_n \vv n + a_1 (i \vv 1) + \ldots + a_n (i \vv n)$. -\end{remark} - -\begin{definition} - Sia $f$ un'applicazione $\RR$-lineare di $V$ spazio vettoriale su $\RR$. Allora - si definisce la \textbf{complessificazione} di $f$, detta $f_\CC : V_\CC \to V_\CC$, - in modo tale che $f_\CC(\v + i \w) = f(\v) + i f(\w)$. -\end{definition} - -\begin{remark} - Si verifica infatti che $f_\CC$ è $\CC$-lineare. - \begin{itemize} - \item $f_\CC((\vv1 + i \ww1) + (\vv2 + i \ww2)) = f_\CC((\vv1 + \vv2) + i (\ww1 + \ww2)) = - f(\vv1 + \vv2) + i f(\ww1 + \ww2) = (f(\vv1) + i f(\ww1)) + (f(\vv2) + i f(\ww2)) = - f_\CC(\vv1 + i\ww1) + f_\CC(\vv2 + i\ww2)$. - - \item $f_\CC((a+bi)(\v + i\w)) = f_\CC(a\v-b\w + i(a\w+b\v)) = f(a\v - b\w) + i f(a\w + b\v) = - af(\v) - bf(\w) + i(af(\w) + bf(\v)) = (a+bi)(f(\v) + if(\w)) = (a+bi) f_\CC(\v + i\w)$. - \end{itemize} -\end{remark} - -\begin{proposition} - Sia $f_\CC$ la complessificazione di $f \in \End(V)$, dove $V$ è uno spazio vettoriale su $\RR$. - Sia inoltre $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Valgono allora i seguenti risultati: - - \begin{enumerate}[(i)] - \item $\restr{(f_\CC)_\RR}{V}$ assume gli stessi valori di $f$, - \item $M_\basis(f_\CC) = M_\basis(f) \in M(n, \RR)$, - \item $M_{\basis \cup i \basis}((f_\CC)_\RR) = \Matrix{M_\basis(f) & \rvline & 0 \\ \hline 0 & \rvline & M_\basis(f)}$. - \end{enumerate} -\end{proposition} - -\begin{proof}Si dimostrano i risultati separatamente. - \begin{enumerate}[(i)] - \item Si osserva che $(f_\CC)_\RR(\vv i) = f_\CC(\vv i) = f(\vv i)$. Dal momento che - $(f_\CC)_\RR$ è $\RR$-lineare, si conclude che $(f_\CC)_\RR$ assume gli stessi valori - di $f$. - - \item Dal momento che $\basis$, nell'identificazione di $(\v, \vec 0)$ come $\v$, è - sempre una base di $V_\CC$, e $f_\CC(\vv i) = f(\vv i)$, chiaramente - $[f_\CC(\vv i)]_\basis = [f(\vv i)]_\basis$, e quindi $M_\basis(f_\CC) = M_\basis(f)$, - dove si osserva anche che $M_\basis(f) \in M(n, \RR)$, essendo $V$ uno spazio vettoriale - su $\RR$. - - \item Sia $f(\vv i) = a_1 \vv 1 + \ldots + a_n \vv n$ con $a_1$, ..., $a_n \in \RR$. Come - osservato in (i), $\restr{(f_\CC)_\RR}{\basis} = \restr{(f_\CC)_\RR}{\basis}$, e quindi - la prima metà di $M_{\basis \cup i \basis}((f_\CC)_\RR)$ è formata da due blocchi: uno - verticale coincidente con $M_\basis(f)$ e un altro completamente nullo, dal momento che - non compare alcun termine di $i \basis$ nella scrittura di $(f_\CC)_\RR(\vv i)$. Al - contrario, per $i \basis$, $(f_\CC)_\RR(i \vv i) = f_\CC(i \vv i) = i f(\vv i) = a_1 (i \vv 1) + - \ldots + a_n (i \vv n)$; pertanto la seconda metà della matrice avrà i due blocchi della prima metà, - benché scambiati. - \end{enumerate} -\end{proof} - -\begin{remark} - Dal momento che $M_\basis(f_\CC) = M_\basis(f)$, $f_\CC$ e $f$ condividono lo stesso polinomio caratteristico - e vale che $\Sp(f) \subseteq \Sp(f_\CC)$, dove vale l'uguaglianza se e solo se tale polinomio caratteristico - è completamente riducibile in $\RR$. Inoltre, se $V_\lambda$ è l'autospazio su $V$ dell'autovalore $\lambda$, l'autospazio - su $V_\CC$, rispetto a $f_\CC$, è invece ${V_\CC}_\lambda = V_\lambda + i V_\lambda$, la cui - dimensione rimane invariata rispetto a $V_\lambda$, ossia $\dim V_\lambda = \dim {V_\CC}_\lambda$ - (infatti, analogamente a prima, una base di $V_\lambda$ può essere identificata come base - anche per ${V_\CC}_\lambda$). -\end{remark} - -\begin{proposition} - Sia $f_\CC$ la complessificazione di $f \in \End(V)$, dove $V$ è uno spazio vettoriale su $\RR$. - Sia inoltre $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$. Allora un endomorfismo - $\tilde g : V_\CC \to V_\CC$ complessifica un endomorfismo $g \in \End(V)$ $\iff$ $M_\basis(\tilde g) \in M(n, \RR)$. -\end{proposition} - -\begin{proof} - Se $\tilde g$ complessifica $g \in \End(V)$, allora, per la proposizione precedente, - $M_\basis(\tilde g) = M_\basis(g) \in M(n, \RR)$. Se invece $A = M_\basis(\tilde g) \in M(n, \RR)$, - si considera $g = M_\basis\inv(A) \in \End(V)$. Si verifica facilemente che $\tilde g$ non è altro che - il complessificato di tale $g$: - - \begin{itemize} - \item $\tilde g (\vv i) = g(\vv i)$, dove l'uguaglianza è data dal confronto delle matrici associate, - e quindi $\restr{\tilde g}{V} = g$; - \item $\tilde g(\v + i\w) = \tilde g(\v) + i \tilde g(\w) = g(\v) + i g(\w)$, da cui la tesi. - \end{itemize} -\end{proof} - -\begin{proposition} - Sia $\varphi$ un prodotto scalare di $V$ spazio vettoriale su $\RR$. Allora esiste un - unico prodotto hermitiano $\varphi_\CC : V_\CC \times V_\CC \to \CC$ che estende $\varphi$ (ossia tale che - $\restr{\varphi_\CC}{V \times V} = \varphi$), il quale assume la stessa segnatura - di $\varphi$. -\end{proposition} - -\begin{proof} - Sia $\basis$ una base di Sylvester per $\varphi$. Si consideri allora il prodotto - $\varphi_\CC$ tale che: - - \[ \varphi_\CC(\vv1 + i\ww1, \vv2 + i\ww2) = \varphi(\vv1, \vv2) + \varphi(\ww1, \ww2) + i(\varphi(\vv1, \ww1) - \varphi(\ww1, \vv2)). \] - - Chiaramente $\restr{\varphi_\CC}{V \times V} = \varphi$. Si verifica allora che $\varphi_\CC$ è hermitiano: - - \begin{itemize} - \item $\varphi_\CC(\v + i\w, (\vv1 + i\ww1) + (\vv2 + i\ww2))$ $= \varphi(\v, \vv1 + \vv2) + \varphi(\w, \ww1 + \ww2)$ $+ i(\varphi(\v, \ww1 + \ww2)$ $- \varphi(\w, \vv1 + \vv2))$ $= [\varphi(\v, \vv1) + \varphi(\w, \ww1) + i(\varphi(\v, \ww1) - \varphi(\w, \vv1))]$ $+ [\varphi(\v, \vv2) + \varphi(\w, \ww2) + i(\varphi(\v, \ww2) - \varphi(\w, \vv2))] = \varphi_\CC(\v + i\w, \vv1 + i\ww1) + - \varphi_\CC(\v + i\w, \vv2 + i\ww2)$ (additività nel secondo argomento), - - \item $\varphi_\CC(\v + i\w, (a+bi)(\vv1 + i\ww1)) = \varphi_\CC(\v + i\w, a\vv1-b\ww1 + i(b\vv1+a\ww1)) = - \varphi(\v, a\vv1-b\ww1) + \varphi(\w, b\vv1+a\ww1) + i(\varphi(\v, b\vv1+a\ww1) - \varphi(\w, a\vv1-b\ww1))= - a\varphi(\v, \vv1) - b\varphi(\v, \ww1) + b\varphi(\w, \vv1) + a\varphi(\w, \ww1) + i(b\varphi(\v, \vv1) + a\varphi(\v, \ww1) - a\varphi(\w, \vv1) + b\varphi(\w, \ww1)) = a(\varphi(\v, \vv1) + \varphi(\w, \ww1)) - - b(\varphi(\v, \ww1) - \varphi(\w, \vv1)) + i(a(\varphi(\v, \ww1) - \varphi(\w, \vv1)) + b(\varphi(\v, \vv1) + \varphi(\w, \ww1))) = (a+bi)(\varphi(\v, \vv1) + \varphi(\w, \ww1) + i(\varphi(\v, \ww1) - \varphi(\w, \vv1))) = (a+bi) \varphi_\CC(\v + \w, \vv1 + i\ww1)$ (omogeneità nel secondo argomento), - - \item $\varphi_\CC(\vv1 + i\ww1, \vv2 + i\ww2) = \varphi(\vv1, \vv2) + \varphi(\ww1, \ww2) + i(\varphi(\vv1, \ww2) - \varphi(\ww1, \vv2)) = \conj{\varphi(\vv1, \vv2) + \varphi(\ww1, \ww2) + i(\varphi(\ww1, \vv2) - \varphi(\vv1, \ww2))} = \conj{\varphi(\vv2, \vv1) + \varphi(\ww2, \ww1) + i(\varphi(\vv2, \ww1) - \varphi(\ww2, \vv1))} = \conj{\varphi_\CC(\vv2 + \ww2, \vv1 + \ww1)}$ (coniugio nello scambio degli argomenti). - \end{itemize} - - Ogni prodotto hermitiano $\tau$ che estende il prodotto scalare $\varphi$ ha la stessa matrice associata nella - base $\basis$, essendo $\tau(\vv i, \vv i) = \varphi(\vv i, \vv i)$ vero per ipotesi. Pertanto $\tau$ è - unico, e vale che $\tau = \varphi_\CC$. Dal momento che $M_\basis(\varphi_\CC) = M_\basis(\varphi)$ è - una matrice di Sylvester, $\varphi_\CC$ mantiene anche la stessa segnatura di $\varphi$. -\end{proof} - -\hr - \begin{theorem} (di rappresentazione di Riesz per il prodotto scalare) Sia $V$ uno spazio vettoriale e sia $\varphi$ un suo prodotto scalare non degenere. Allora per ogni $f \in V^*$ esiste un unico $\v \in V$ tale che @@ -601,8 +280,8 @@ \begin{remark}\nl Si possono classificare in modo semplice alcuni di questi gruppi unitari. - \li $A \in U_n \implies 1 = \det(I_n) = \det(A A^*) = \det(A) \conj{\det(A)} = \abs{\det(A)}^2 = 1$. - \li $A = (a) \in U_1 \iff A^* A = I_1 \iff \abs{a}^2 = 1 \iff a = e^{i\theta}$, $\theta \in [0, 2\pi)$, ossia il numero complesso $a$ appartiene alla circonferenza di raggio unitario. + \li $A \in U_n \implies 1 = \det(I_n) = \det(A A^*) = \det(A) \conj{\det(A)} = \abs{\det(A)}^2 = 1$. \\ + \li $A = (a) \in U_1 \iff A^* A = I_1 \iff \abs{a}^2 = 1 \iff a = e^{i\theta}$, $\theta \in [0, 2\pi)$, ossia il numero complesso $a$ appartiene alla circonferenza di raggio unitario. \\ \li $A = \Matrix{a & b \\ c & d} \in SU_2 \iff A A^* = \Matrix{\abs{a}^2 + \abs{b}^2 & a\conj c + b \conj d \\ \conj a c + \conj b d & \abs{c}^2 + \abs{d}^2} = I_2$, $\det(A) = 1$, ossia se il seguente sistema di equazioni è soddisfatto: 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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