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 bcfbf36..f23625d 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 @@ -788,7 +788,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni di segno $\iff d_i > 0$ se $i$ è pari e $d_i < 0$ se $i$ è dispari $\iff (-1)^i \, d_i > 0$, $\forall 1 \leq i \leq n$. \end{proof} -\subsubsection{Sottospazi isotropi e indice di Witt} +\section{Sottospazi isotropi e indice di Witt} \begin{definition}[sottospazio isotropo] Sia $W$ un sottospazio di $V$. Allora si dice che $W$ è un \textbf{sottospazio isotropo} di $V$ diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf index bc92f89..de0acd4 100644 Binary files a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf and b/Geometria 1/4. 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Un insieme linearmente indipendente di $V$ con esattamente $\dim V$ elementi è una base di $V$. Analogamente, un insieme generatore di $V$ con esattamente - $\dim V$ elementi è una base di $V$. + $\dim V$ elementi è una base di $V$. In generale, quando il campo su + cui fonda lo spazio vettoriale è ambiguo, si scrive $\dim_\KK V$ o $[V : \KK]$ + per indicarne la dimensione relativa al campo $\KK$ (per esempio un $\CC$-spazio è + compatibilmente anche un $\RR$-spazio). Sia $\basis = \{\vec{v_1}, \ldots, \vec{v_n}\}$ una base ordinata dello spazio vettoriale $V$. @@ -233,7 +236,8 @@ $\Ker \dual{\vec{t}}$), \item un iperpiano $\Pi$, rappresentato da un'equazione cartesiana $\alpha_1 x_1 + \ldots + \alpha_n x_n = 0$, è esattamente il sottospazio ortogonale a $\Span(\alpha_1, \ldots, \alpha_n)^\perp$ tramite il prodotto scalare standard, \item in generale, un sistema di equazioni omogenee è l'intersezione di più - sottospazi ortogonali. + sottospazi ortogonali, + \item se $\mathbb{F}$ è un'estensione di campo di $\KK$, allora vale $[V : \KK] = [V : \mathbb{F}] [\mathbb{F} : \KK]$ (\textit{teorema delle torri algebriche}). \end{itemize} @@ -464,6 +468,16 @@ \det(\lambda I_n - A)$ (\textit{teorema di Hamilton-Cayley}). + Si elencano adesso i tipi principali di matrici: + + \begin{itemize} + \item $A$ è simmetrica $\defiff A^\top = A$, + \item $A$ è antisimmetrica $\defiff A^\top = -A$, + \item $A$ è hermitiana $\defiff A^* := \left(\overline{A\,}\right)^\top = A$, + \item $A$ è ortogonale ($A \in O(n)$ o $O_n$) $\defiff AA^\top = A^\top A = I_n$, + \item $A$ è unitaria ($A \in U(n)$ o $U_n$) $\defiff AA^* = A^*A = I_n$. + \end{itemize} + \subsection{Rango di una matrice} Si definisce rango di una matrice $A$ il numero di colonne linearmente @@ -527,6 +541,10 @@ \dim \Span(A^1, \ldots, A^n) \iff \dim \Im (A \mid B) = \dim \Im A \iff \rg (A \mid B) = \rg (A)$ (\textit{teorema di Rouché-Capelli}), + \item Data un'applicazione lineare $f : \KK^n \to \KK^m$ determinata + dalla matrice $M$, il sistema di equazioni cartesiane che rappresenta + $\Im f$ si ottiene imponendo la validità del teorema di Rouché-Capelli + sul vettore $\vec x = (x_1, \ldots, x_m)$, ossia imponendo $\rg(M) = \rg(M \mid \vec x)$, \item $A\vec{x} = B$, se la ammette, ha un'unica soluzione se e solo se $\Ker A = \zerovecset \iff \rg A = n$. \end{itemize} @@ -764,7 +782,7 @@ \[ \Ann_{\mathcal{L}(V, W)}(U) = \left\{ f \in \mathcal{L}(V, W) \mid f(U) = \zerovecset \right\}. \] Se $V$ è a dimensione finita, la dimensione di - $\Ann(U)$ è pari a $(\dim V - \dim U) \cdot \dim W$ (è sufficiente + $\Ann_{\mathcal{L}(V, W)}(U)$ è pari a $(\dim V - \dim U) \cdot \dim W$ (è sufficiente prendere una base di $U$, completarla a base di $V$ e notare che $f(U) = \zerovecset \iff$ ogni valutazione in $f$ degli elementi della base di $U$ è nullo $\iff$ la matrice @@ -1719,6 +1737,9 @@ \item $\CI(\restr{\varphi}{U}) = \CI(\varphi) \cap U$, \end{itemize} + Se $U$ è un sottospazio di $V$, $\varphi$ induce un prodotto scalare $\hat \varphi : V/U \times V/U \times \KK$ tale che $\hat \varphi([\vv 1]_U, [\vv 2]_U) = \varphi(\vv 1, \vv 2)$ se e solo se $U \subseteq V^\perp$. In particolare, se $U = V^\perp$, + $\hat \varphi$ è anche non degenere. + Due esempi classici di prodotto scalare sono $\varphi(A, B) = \tr(AB)$ e $\psi(A, B) = \tr(AB^\top)$, entrambi su $M(n, \KK)$. I due prodotti sono entrambi non degeneri, e vale che: @@ -1793,8 +1814,10 @@ una base ortogonale e se ne normalizzino i vettori anisotropi, facendo infine eventuali considerazioni dimensionali per dimostrare la seconda parte dell'enunciato), - \item $\varphi > 0 \iff \sigma = (n, 0, 0)$ e $\varphi < 0 \iff \sigma = (0, n, 0)$, - \item $\varphi \geq 0 \iff \sigma = (n - k, 0, k)$ e $\varphi \leq 0 \iff \sigma = (0, n - k, k)$, + \item $\varphi > 0 \iff \sigma = (n, 0, 0)$, + \item $\varphi < 0 \iff \sigma = (0, n, 0)$, + \item $\varphi \geq 0 \iff \sigma = (n - k, 0, k)$, + \item $\varphi \leq 0 \iff \sigma = (0, n - k, k)$, con $0 \leq k \leq n$ tale che $k = \dim V^\perp$, \item I vettori isotropi di una base ortogonale sono una base di $V^\perp$, @@ -1826,31 +1849,66 @@ un piano iperbolico ed esistono $\lambda_1$, $\lambda_2 \in \RR$ tali per cui $\lambda_1 \v + \lambda_2 \w$ è isotropo. + \subsection{Prodotto hermitiano} + + Sia $V$ un $\CC$-spazio. Allora una mappa $\varphi : V \times V \to \CC$ si + dice prodotto hermitiano (e quindi si dice che $\varphi \in \PH(V)$, l'$\RR$-spazio dei + prodotti hermitiani\footnote{Infatti, se $\lambda \in \CC \setminus \RR$ e $\varphi \in \PH(V)$, $\lambda \varphi$ \underline{non} è un prodotto hermitiano, mancando della proprietà del coniugio.}) se è una forma sesquilineare, ossia se è antilineare + nel primo argomento ed è lineare nel secondo\footnote{In realtà questa convenzione è spesso e volentieri implementata nelle ricerche di Fisica, mentre in Matematica si tende in realtà a mettere l'antilinearità nel secondo argomento. Il corso ha comunque implementato la prima delle due convenzioni, e così si è riportato in queste schede la convenzione scelta.}, e se il coniugio applicato a $\varphi$ ne inverte gli argomenti. In particolare $\varphi$ è un prodotto hermitiano se: + + \begin{enumerate}[(i)] + \item $\varphi(\v, \lambda \U + \w) = \lambda \varphi(\v, \U) + \varphi(\v, \w)$, + $\forall \v$, $\U$, $\w \in V$, $\lambda \in \KK$, + \item $\conj{\varphi(\v, \w)} = \varphi(\w, \v)$. + \end{enumerate} + + Un prodotto hermitiano $\varphi$ si comporta pressoché come un prodotto + scalare su $\RR$. Infatti tale prodotto soddisfa le seguenti proprietà: + + \begin{itemize} + \item + \end{itemize} + \subsubsection{Funzionali rappresentabili} Un funzionale $f \in \dual V$ si dice rappresentabile tramite $\varphi$ se $f \in \Im \alpha_\varphi$, ossia se $\exists \v \in V \mid f = \varphi(\v, \cdot)$. - Dal momento che $\Im(\alpha_\varphi) = \Ann(V^\perp)$, $f$ è rappresentabile + Dal momento che $\Im(\alpha_\varphi) = \Ann(V^\perp)$ (l'inclusione verso destra è facile da dimostrare e l'uguaglianza è data dall'uguaglianza dimensione), $f$ è rappresentabile se e solo se $V^\perp \subseteq \Ker f$. - Se $\varphi$ è non degenere, ogni funzionale $f$ è rappresentabile in modo unico (teorema - di rappresentazione di Riesz; infatti $\alpha_\varphi$ sarebbe in tal caso - un isomorfismo). + Se $\varphi$ è non degenere, ogni funzionale $f$ è rappresentabile in modo unico (\textit{teorema + di rappresentazione di Riesz}; infatti $\alpha_\varphi$ sarebbe in tal caso + un isomorfismo). In particolare, se $\varphi$ è un prodotto scalare, tale + vettore $\v$, data una base ortogonale $\basis = \{ \vv 1, \ldots, \vv n\}$, è + determinato nel seguente modo: + + \[ \v = \sum_{i=1}^n \frac{f(\vv i)}{\varphi(\vv i, \vv i)} \, \vv i. \] - In particolare, $f$ è rappresentabile se e solo se, scelta una base $\basis$ di + Se invece $\varphi$ è un prodotto hermitiano, tale vettore $\v$ si determina + nel seguente altro modo: + + \[ \v = \sum_{i=1}^n \conj{\left(\frac{f(\vv i)}{\varphi(\vv i, \vv i)}\right)} \, \vv i. \] + + In generale, $f$ è rappresentabile se e solo se, scelta una base $\basis$ di $V$, il sistema $M_\basis(\varphi) \x = [f]_{\dual \basis}$ è risolvibile. Se $W$ è un supplementare di $V^\perp$, e dunque $V = W \oplus V^\perp$, allora - $\restr{\varphi}{W}$ è non degenere, e dunque $\alpha_{\restr{\varphi}{W}}$ è + $\restr{\varphi}{W}$ è non degenere, e dunque $\restr{\alpha_{\varphi}}{W} : W \to \Im \alpha_\varphi$ è un isomorfismo da $W$ a $\Im \alpha_\varphi$ (quindi se $f$ è rappresentabile, lo è tramite un unico vettore di $W$). In particolare, se $\v$ rappresenta $f$, allora $\Ker f = \v^\perp$; da cui - segue che $\v \in (\Ker f)^\perp = \Span(\v) + V^\perp$. Inoltre - $\dim (\Ker f)^\perp = \dim V - \dim \Ker f + \dim (\Ker f \cap V^\perp) - = \dim V - \dim \Ker f + \dim V^\perp$. Se $f$ non è l'applicazione nulla, - $\v \notin \CI(\varphi)$, e quindi $\dim (\Ker f)^\perp = \dim V^\perp + 1$, da - cui segue che $\Ker f^\perp = \Span(\v) \oplus^\perp V^\perp$. + segue che $(\Ker f)^\perp = \v^\dperp = \Span(\v) + V^\perp$. Se $f$ non è + l'applicazione nulla, $\vec v \notin V^\perp$, e quindi $\Span(\v) \cap V^\perp = \zerovecset \implies (\Ker f)^\perp = \Span(\v) \oplus^\perp V^\perp$. Quindi, + per computare un vettore $\vv 0$ che rappresenti $f$ è sufficiente prendere + un supplementare $\Span(\U)$ di $V^\perp$ in $(\Ker f)^\perp$ (infatti l'aggiunta + di un vettore di $V^\perp$ non varierebbe l'immagine secondo $\alpha_\varphi$) e + computare $\lambda \in \KK \mid \vv 0 = \lambda \U$ nel seguente + modo: + + \[ \lambda = \frac{\varphi(\lambda \U, \w)}{\varphi(\U, \w)} = \frac{f(\w)}{\varphi(\U, \w)}, \] + + dove $\w \notin \Ker f$. \subsubsection{Algoritmo di ortogonalizzazione di Gram-Schmidt} @@ -1896,13 +1954,13 @@ \begin{itemize} \item $V^\perp$ è un sottospazio isotropo, - \item Se $W$ è isotropo, allora $\dim W \leq \frac{\dim V + \dim \Rad(\varphi)}{2}$, - \item Se $W$ è isotropo e $\varphi$ è non degenere, allora $\dim W \leq \frac{1}{2} \dim V$, + \item Se $W$ è isotropo, allora $\dim W \leq \lfloor \frac{\dim V + \dim \Rad(\varphi)}{2} \rfloor$, + \item Se $W$ è isotropo e $\varphi$ è non degenere, allora $\dim W \leq \lfloor \frac{1}{2} \dim V \rfloor$, \item Se $\KK = \RR$, allora $W(\varphi) = \min\{ i_+, i_- \} + i_0$ (è sufficiente considerare - una base di Sylvester e creare una nuova base i cui i vettori sono o isotropi o della forma $\vv i - \ww i$, dove $q(\vv i) = 1$ e $q(\ww i) = 1$, concludendo con discussioni dimensionali), - \item Se $\KK = \CC$, allora $W(\varphi) = \lfloor \frac{\dim V + \dim V^\perp}{2} \rfloor$, - \item Se $\varphi$ è definito, allora $W(\varphi) = 0$, - \item Se $\varphi$ è semidefinito, allora $W(\varphi) = i_0$ (e $W = V^\perp$ è un sottospazio + una base di Sylvester e costruire un nuovo insieme linearmente indipendente $\basis_W$ i cui i vettori sono o isotropi o della forma $\vv i - \ww i$, dove $q(\vv i) = 1$ e $q(\ww i) = 1$, mostrando che $W = \Span(\basis_W)$ è isotropo, concludendo con discussioni dimensionali), + \item Se $\KK = \CC$, allora $W(\varphi) = \lfloor \frac{\dim V + \dim V^\perp}{2} \rfloor$ (è sufficiente considerare una base di Sylvester per $\varphi$, costruire un nuovo insieme linearmente indipendente $\basis_W$ prendendo quante più coppie $(\vv i, \vv j)$ possibili di vettori della base non isotropi poi associate al vettore $\vv i + i \vv j$, mostrando infine che $W = \Span(\basis_W)$ è isotropo e che è sicuramente massimale perché realizza la dimensione massima possibile secondo le precedenti proposizioni), + \item Se $\KK=\RR$ e $\varphi$ è definito, allora $W(\varphi) = 0$, + \item Se $\KK=\RR$ e $\varphi$ è semidefinito, allora $W(\varphi) = i_0$ (e $W = V^\perp$ è un sottospazio isotropo di tale dimensione). \end{itemize} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 5ca3f35..0449631 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -128,6 +128,7 @@ \newcommand{\conj}[1]{\overline{#1}} +\DeclareMathOperator{\PH}{PH} \DeclareMathOperator{\PS}{PS} \let\imm\Im