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coordinate della rappresentazione della base (infatti ogni iperpiano è il kernel di un funzionale $f \in \dual{V}$, e $M^\basis_{1_\KK}(f) \, [\vec{v}]_\basis = 0$ è l'equazione cartesiana; è sufficiente prendere una base di $\Pi$ e completarla a base di $V$ con un vettore $\vec{t}$, considerando infine - $\Ker \dual{\vec{t}}$). + $\Ker \dual{\vec{t}}$), + \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. \end{itemize} @@ -823,12 +826,10 @@ di $V_i$, dove $k_i = \dim V_i$. Si definisce l'applicazione $n$-lineare $\dual{\vec{v^{(1)}_{j_1}}} \otimes \cdots \otimes \dual{\vec{v^{(n)}_{j_n}}}\in \Mult(V_1 \times \ldots - \times V_n, \KK)$ univocamente determinata dalla relazione: - + \times V_n, \KK)$ univocamente determinata dalla seguente relazione: \[ \dual{\vec{v^{(1)}_{j_1}}} \otimes \cdots \otimes \dual{\vec{v^{(n)}_{j_n}}}(\vec{w_1}, \ldots, \vec{w_n}) = \dual{\vec{v^{(1)}_{j_1}}}(\vec{w_1}) \cdots \dual{\vec{v^{(n)}_{j_n}}}(\vec{w_n}). \] Si definisce l'insieme $\basis_{\otimes}$ nel seguente modo: - \[ \basis_{\otimes} = \left\{ \dual{\vec{v^{(1)}_{j_1}}} \otimes \cdots \otimes \dual{\vec{v^{(n)}_{j_n}}} \mid 1 \leq j_1 \leq k_1, \, \ldots, \, 1 \leq j_n \leq k_n \right\}. \] Poiché ogni applicazione $n$-lineare è univocamente determinata @@ -862,12 +863,10 @@ si definisce il prodotto simmetrico (o \textit{prodotto vee}) $\dual{\vec{v_{i_1}}} \vee \cdots \vee \dual{\vec{v_{i_k}}}$ tra elementi della base come la forma $k$-lineare simmetrica - determinata dalla relazione: - + determinata dalla seguente relazione: \[ \dual{\vec{v_{i_1}}} \vee \cdots \vee \dual{\vec{v_{i_k}}} = \sum_{\sigma \in S_k} \dual{\vec{v_{i_{\sigma(1)}}}} \otimes \cdots \otimes \dual{\vec{v_{i_{\sigma(k)}}}}. \] Si definisce l'insieme: - \[\basis_{\Sym} = \left\{ \dual{\vec{v_{i_1}}} \vee \cdots \vee \dual{\vec{v_{i_k}}} \mid 1 \leq i_1 \leq \cdots \leq i_k \leq n \right\}. \] L'insieme $\basis_{\Sym}$ è sia generatore che linearmente @@ -1554,7 +1553,16 @@ a $\conj{\lambda}$ coniugando una base dell'autospazio generalizzato relativo a $\lambda$ (in particolare i due spazi hanno la stessa dimensione), \item Se $\KK=\CC$, la forma canonica di Jordan contiene tanti blocchi di taglia $t$ - relativi a $\lambda$ quanti ve ne sono di relativi a $\conj{\lambda}$. + relativi a $\lambda$ quanti ve ne sono di relativi a $\conj{\lambda}$, + \item Esistono e sono unici i due endomorfismi $\mu$, $\delta \in \End(V)$ + tale che $\mu$ sia diagonalizzabile, $\delta$ sia nilpotente e che + $f = \mu + \delta$ (se esiste la forma canonica di Jordan), + \item Se $\forall \lambda \in \Sp(f)$, $\mu_g(\lambda) = 1$, allora + esiste un numero finito di sottospazi invarianti e sono tutte le possibili + somme dirette dei sottospazi degli autospazi generalizzati, %TODO: migliorare + \item Se $\KK$ è infinito ed esiste $\lambda \in \Sp(f)$ tale per cui + $\mu_g(\lambda) > 1$, allora esiste un numero infinito di sottospazi invarianti + per ogni dimensione, da $1$ a $\dim V -1$. %TODO: migliorare \end{itemize} \subsubsection{Calcolo di una base di Jordan} @@ -1667,6 +1675,13 @@ Si dice che $\varphi$ è definito se è definito positivo o definito negativo. Analogamente $\varphi$ è semidefinito se è semidefinito positivo o semidefinito negativo. + Si scrive $\v^\perp$ per indicare tutti i vettori ortogonali a $\v$ (e quindi + $\v^\perp = \Span(\v)^\perp$). Si definisce $\iota : W \to V$ come l'applicazione + tale per cui $\iota(\w) = \w$. Si scrive $\restr{\varphi}{U}$ con $U$ sottospazio + di $V$ per indicare il prodotto scalare $\restr{\varphi}{U \times U}$. + + Sia ora $V$ di dimensione finita. + \begin{itemize} \item $M_\basis(\varphi)$ è simmetrica, \item $\varphi(\v, \w) = [\v]_\basis^\top M_\basis(\varphi) [\w]_\basis$, @@ -1676,15 +1691,43 @@ \item $W^\perp = \Ker i^\top \circ a_\varphi$, \item $a_\varphi(W^\perp) = \Ann(W) \cap \Im a_\varphi$, \item $\dim W + \dim W^\perp = \dim V + \dim (W \cap V^\perp)$ (da sopra), - \item $V = W \oplus^\perp W^\perp$ se $\restr{\varphi}{W}$ è non degenere ($\iff W \cap W^\perp = \Rad(\restr{\varphi}{W}) = \zerovecset$), + \item $V = W \oplus^\perp W^\perp$ se e solo se $\restr{\varphi}{W}$ è non degenere ($\iff W \cap W^\perp = \Rad(\restr{\varphi}{W}) = \zerovecset$), + \item $V = W + W^\perp$ se e solo se $\Rad(\restr{\varphi}{W}) \subseteq \Rad(\varphi)$, + \item se $V = U \oplus^\perp W$, allora $\Rad(\varphi) = \Rad(\restr{\varphi}{U}) \oplus \Rad(\restr{\varphi}{W})$, + \item $V = \Span(\w) \oplus^\perp \Span(\w)^\perp \iff q(\w) \neq 0 \iff \w \notin \CI(\varphi)$, + \item $\Span(\w) \subseteq \Span(\w)^\perp \iff \w \in \CI(\varphi)$, \item $(W^\perp)^\perp = W^\dperp = W + \Rad(\varphi) = W + V^\perp$, + \item $\v \in V^\perp \iff \Span(\v)^\perp = V$, + \item $W^\perp = (\Span(W))^\perp$, + \item $W^\perp = \bigcap_{\w \in U} \w^\perp$, + \item se $\ww 1$, ..., $\ww k$ generano $W$, allora $W^\perp = \bigcap_{i = 1}^k \ww i ^\perp$, + \item $W^\perp = \alpha_\varphi\inv(\Ann(W))$, + \item $\alpha_\varphi(W^\perp) = \Ann(W) \cap \Im \alpha_\varphi$, + \item $V \subseteq W \implies W^\perp \subseteq V^\perp$, \item $(U + W)^\perp = U^\perp \cap W^\perp$, + \item $((W^\perp)^\perp)^\perp = (W + V^\perp)^\perp = W^\perp \cap (V^\perp)^\perp = W^\perp \cap V = W^\perp$, \item $(U \cap W)^\perp \supseteq U^\perp + W^\perp$, \item $(U \cap W)^\perp = U^\perp + W^\perp$, se $\varphi$ è non degenere, \item $\varphi$ è definito $\iff$ $\CI(\varphi) = \zerovecset$, + \item $\varphi$ è (semi)definito $\implies$ ogni sua restrizione è (semi)definita, \item $\varphi$ è semidefinito $\iff$ $\CI(\varphi) = V^\perp = \Rad(\varphi)$ (considera l'esistenza di due vettori $\v$, $\w \in V$ con forme quadratiche discordi, osserva che sono linearmente indipendenti - e trova un $\lambda \in \KK$ tale per cui $\v + \lambda \w$ crea un assurdo). + e trova un $\lambda \in \KK$ tale per cui $\v + \lambda \w$ crea un assurdo), + \item $\Im(\alpha_\varphi) \subseteq \Ann(V^\perp)$ (se $V$ è di dimensione infinita), + \item $\Im(\alpha_\varphi) = \Ann(V^\perp)$ (se $V$ è di dimensione finita), + \item $\Rad(\restr{\varphi}{U}) = U^\perp \cap U$, + \item $\CI(\restr{\varphi}{U}) = \CI(\varphi) \cap U$, + \end{itemize} + + 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: + + \begin{itemize} + \item $\restr{\varphi}{\Sym(n, \KK)} = \restr{\psi}{\Sym(n, \KK)}$, + \item $\restr{\varphi}{\Lambda(n, \KK)} = -\restr{\psi}{\Lambda(n, \KK)}$, + \item se $\Char \KK \neq 2$, $V = \Sym(n, \KK) \oplus^\perp \Lambda(n, \KK)$, + per ambo i prodotti scalari. \end{itemize} Se $\basis'$ è un'altra base di $V$, vale il seguente \textit{teorema di cambiamento di base}: @@ -1775,6 +1818,40 @@ Sylvester. Per il teorema di Sylvester, tale base esiste sempre, e la matrice di Sylvester è unica per ogni prodotto scalare $\varphi$. + Se $M \in \Sym(2, \RR)$, $\det(M) < 0 \iff \sigma(M) = (1, 1, 0)$ (e dunque + se e solo se $M$ rappresenta un piano iperbolico). Al contrario $\det(M) > 0$ + se e solo se $M$ è definita (e in tal caso è definita positiva se il suo + primo elemento è positivo, e negativa se è negativo). Se $\KK=\RR$, $q(\v) > 0$ e $q(\w) < 0$, + allora $\v$ e $\w$ sono linearmente indipendenti; in particolare $\Span(\v, \w)$ è + un piano iperbolico ed esistono $\lambda_1$, $\lambda_2 \in \RR$ tali per + cui $\lambda_1 \v + \lambda_2 \w$ è isotropo. + + \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 + 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). + + In particolare, $f$ 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 + 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 $\Ker f^\perp = \Span(\v) \oplus^\perp V^\perp$. + \subsubsection{Algoritmo di ortogonalizzazione di Gram-Schmidt} Data una base $\basis$ di $V$, se $\abs{\CI(\varphi) \cap \basis} \leq 1$ (ossia se ogni vettore di @@ -1823,6 +1900,7 @@ \item Se $W$ è isotropo e $\varphi$ è non degenere, allora $\dim W \leq \frac{1}{2} \dim V$, \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 isotropo di tale dimensione). @@ -1853,6 +1931,80 @@ rango, allora $V$ e $W$ sono isometrici tra loro (come conseguenza stavolta del teorema di Sylvester complesso). + \subsection{Teorema spettrale reale e complesso} + + Se $f$ è simmetrico o hermitiano, esiste sempre una base ortonormale di autovettori + per $f$. $f$ è normale se e solo se è diagonalizzabile. Esiste sempre $S \in \Sym(n, \RR)$ tale per cui $S^2 = A$, con $A \in \Sym(n, \RR)$. Se $S$ è definita positiva, + tale $S$ è unica. Se $A \in \GL(n, \RR)$, esistono e sono unici $P \in O(n)$, + $S \in \Sym(n, \RR)$ tali per cui $A = PS$ (in particolare $S = \sqrt{A A^\top}$). + + Se $\varphi$ è definito positivo e $\psi$ è un altro prodotto scalare, allora + i due prodotti sono simultaneamente ortogonalizzabili. È sufficiente prendere + una base $\basis$ ortonormale di $\varphi$, e trovare la base ortonormale $\basis'$ di autovettori + che rende $M_\basis(\psi)$ diagonale. In tale base $\basis'$, $M_{\basis'}(\varphi)$ + è l'identità e $M_\basis(\psi)$ è diagonale: dunque la base è ortogonale per ambo + i prodotti scalari. + + \subsection{Complementi sugli spazi affini} + + \begin{itemize} + \item se $A \cap B \neq \emptyset$, vale la formula di Grassmann, + \item se $A \cap B = \emptyset$, $\dim (A + B) = \dim A + \dim B - \dim (A_0 \cap B_0) + 1$, + \item ogni affinità $f(\x) = M \x + \vec t$ può scriversi in forma matriciale + come: + + \[ \Matrix{M & \vec t \\ 0 & 1}. \] + \end{itemize} + + La matrice $\MM(p \circ f)$, con $f(\x) = M\x + \vec t$, varia nel seguente + modo: + \begin{gather*} + \tiny + \MM(p \circ f) = {\hat M}^\top \MM(p) \hat M = \\ \Matrix{M^\top \AA(p) M & \rvline & M^\top(\AA(p) \vec t + \Ll(p)) \, \\ \hline \, \left(M^\top(\AA(p) \vec t + \Ll(p))\right)^\top & \rvline & p(\vec t)}, + \end{gather*} + + Una conica è a centro se e solo se è risolvibile il sistema $(\AA(p) \vec t + \Ll(p)$. + + Sia $\KK=\CC$. Allora ogni conica è affinemente equivalente ad + un'equazione canonica della seguente tabella, unicamente + determinata dagli invarianti $\rg(\MM(p))$ e $\rg(\AA(p))$. + + \begin{center} + \tiny + \begin{tabular}{|l|l|l|l|l|} + \hline + & $\rg(\MM(p))$ & $\rg(\AA(p))$ & Eq.~canonica & A centro \\ \hline + $\mathcal{C}_1$ & 3 & 2 & $x^2+y^2=1$ & Sì \\ \hline + $\mathcal{C}_2$ & 3 & 1 & $x^2=y$ & No \\ \hline + $\mathcal{C}_3$ & 2 & 2 & $x^2+y^2=0$ & Sì \\ \hline + $\mathcal{C}_4$ & 2 & 1 & $x^2=1$ & Sì \\ \hline + $\mathcal{C}_5$ & 1 & 1 & $x^2=0$ & Sì \\ \hline + \end{tabular} + \end{center} + + Sia $\KK=\RR$. Allora ogni conica è affinemente equivalente ad + un'equazione canonica della seguente tabella, unicamente + determinata dagli invarianti $\rg(\MM(p))$, $\rg(\AA(p))$, + $S(\MM(p)) := \abs{\iota_+(\MM(p)) - \iota_-(\MM(p))}$ e + $S(\AA(p)) := \abs{\iota_+(\AA(p)) - \iota_-(\AA(p))}$. \\[0.1in] + + \begin{center} + \tiny + \begin{tabular}{|l|l|l|l|l|l|} + \hline + & $\rg(\MM(p))$ & $\rg(\AA(p))$ & $S(\MM(p))$ & $S(\AA(p))$ & Eq.~canonica \\ \hline + $\mathcal{C}_1$ & 3 & 2 & 1 & 2 & $x^2+y^2-1=0$ \\ \hline + $\mathcal{C}_2$ & 3 & 2 & 1 & 0 & $x^2-y^2-1=0$ \\ \hline + $\mathcal{C}_3$ & 3 & 1 & 1 & 1 & $x^2-y=0$ \\ \hline + $\mathcal{C}_4$ & 2 & 2 & 0 & 0 & $x^2-y^2=0$ \\ \hline + $\mathcal{C}_5$ & 2 & 1 & 0 & 1 & $x^2-1=0$ \\ \hline + $\mathcal{C}_6$ & 1 & 1 & 1 & 1 & $x^2=0$ \\ \hline + $\mathcal{C}_7$ & 3 & 2 & 3 & 2 & $x^2+y^2+1=0$ \\ \hline + $\mathcal{C}_8$ & 2 & 2 & 2 & 2 & $x^2+y^2=0$ \\ \hline + $\mathcal{C}_9$ & 2 & 1 & 2 & 1 & $x^2+1=0$ \\ \hline + \end{tabular} + \end{center} 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