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z1+)3h&XWO9En+NT2`_{qX-SCWxAF*UqO|)5V9GpA7aNkrb0aWIfq*5333gv*cS69p zd0H)I$S`1}r3}LavVxYCcw`M31Gf?dMZ&+ zaCfLLWb=!);J#gAOeoim9bnAPeuiOZXFp)c9U1F~9VV8QfuFEWVhgqs34PGF_e6|Y z3}4GS3At7&G8xLmvM&`>OzeNIuonx4icPpRWZZ5lSo(Q#Ljh4^=%rOBo6 z++~MOp*rm_DZ!&8H2Gg>!FalpiFH|t#7@W%x(>I7RgbhP-$U z>+Wi6Uz_Q|rx(ub=Jwn1A*%U&y9u9^;5XE^dB$DPv8EGuur<1y=UhLhvA(`xuEN&H z-3EA}5iyt-xb<_DKTmNbWu22mxx~h4b?~v?`tG)_?jLt$)|5jv83yH3r#38UEdL+< Cri3>D diff --git a/Geometria 1/Scheda riassuntiva/main.tex b/Geometria 1/Scheda riassuntiva/main.tex index e946773..b464f66 100644 --- a/Geometria 1/Scheda riassuntiva/main.tex +++ b/Geometria 1/Scheda riassuntiva/main.tex @@ -1,13 +1,12 @@ \documentclass[10pt,landscape]{article} \usepackage{amssymb,amsmath,amsthm,amsfonts} -\usepackage{personal_commands} \usepackage{multicol,multirow} \usepackage{marvosym} \usepackage{calc} \usepackage{ifthen} \usepackage[landscape]{geometry} \usepackage[colorlinks=true,citecolor=blue,linkcolor=blue]{hyperref} - +\usepackage{personal_commands} \ifthenelse{\lengthtest { \paperwidth = 11in}} { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } @@ -1193,10 +1192,199 @@ autovettori in un'unica base $\basis$ di $V$, si otterrà dunque che una base in cui le matrici associate di $f$ e $g$ sono diagonali. - %\item vale sempre che $p_f(f) = 0$ (teorema di Hamilton-Cayley -- - % data una matrice associata $A$ di $f$, è sufficiente studiare - % l'identità - % $(A-\lambda I_n) \cdot \adj(A-\lambda I_n) = p_f(\lambda) I_n$) + \subsection{Prodotto scalare e congruenza} + Si consideri una mappa $\varphi : V \times V \to \KK$. Si dice che + $\varphi$ è un prodotto scalare (e quindi che $\varphi \in \PS(V)$, lo spazio dei prodotti scalari) se è una forma bilineare simmetrica. + In particolare vale la seguente identità: + + \[ \varphi\left( \sum_{i=1}^s a_i \vv i, \sum_{j=1}^t b_j \ww j \right) = + \sum_{i=1}^s \sum_{j=1}^t a_i b_j \varphi(\vv i, \ww j). \] + + Se $\basis = \{ \vv 1, \ldots ,\vv n \}$ è una base di $V$, si definisce $M_\basis(\varphi) = (\varphi(\vv i, \vv j))_{i,j=1\mbox{--}n}$ come la matrice associata al prodotto scalare $\varphi$. In particolare, + se $a_\varphi : V \to V^*$ è la mappa lineare che associa a $\v$ il funzionale $\varphi(\v, \cdot) \in V^*$ + tale che $\varphi(\v, \cdot)(\w) = \varphi(\v, \w)$. + + Si definisce prodotto scalare \textit{standard} il prodotto $\varphi$ tale che + $\varphi(\v, \w) = [\v]_\basis^\top [\w]_\basis$. + + Si dice che due vettori $\v$, $\w \in V$ sono ortogonali tra loro, scritto come $\v \perp \w$, se + $\varphi(\v, \w) = 0$. Dato $W$ sottospazio di $V$, si definisce $W^\perp$ come il sottospazio di $V$ dei vettori ortogonali a tutti i vettori di $W$. Si dice che $\varphi$ è non degenere se $V^\perp = \zerovecset$. + Si scrive in particolare che $V^\perp = \Rad(\varphi)$. + + Si dice che $V = U \oplus^\perp W$ (ossia che $U$ e $W$ sono in somma diretta ortogonale) se $V = U \oplus W$ e $U \subseteq W^\perp$. Sia $i : W \to V$ tale che $\w \mapsto \w$. Si scrive $\restr{\varphi}{W}$ intendendo $\restr{\varphi}{W \times W}$. + + Ad ogni prodotto scalare si può associare una forma quadratica (e viceversa) $q : V \to \KK$ tale che + $q(\v) = \varphi(\v, \v)$. Un vettore $\v \in V$ si dice isotropo se $q(\v) = 0$ (altrimenti si dice + anisotropo). Si definisce il cono isotropo $\CI(\varphi)$ come l'insieme dei vettori isotropi di $V$. + + Se $\KK = \RR$, si dice che $\varphi$ è semidefinito positivo ($\varphi \geq 0$) se $q(\v) \geq 0$ $\forall \v \in V$, e che è semidefinito negativo ($\varphi \leq 0$) se $q(\v) \leq 0$ $\forall \v \in V$. Si dice + che $\varphi$ è definito positivo ($\varphi > 0$) se $\varphi \geq 0$ e se $q(\v) = 0 \iff \v = \vec 0$, + e che è definito negativo ($\varphi < 0$) se $\varphi \leq 0$ e se $q(\v) = 0 \iff \v = \vec 0$. + + Si dice che $\varphi$ è definito se è definito positivo o definito negativo. Analogamente $\varphi$ + è semidefinito se è semidefinito positivo o semidefinito negativo. + + \begin{itemize} + \item $M_\basis(\varphi)$ è simmetrica, + \item $\varphi(\v, \w) = [\v]_\basis^\top M_\basis(\varphi) [\w]_\basis$, + \item $M_\basis(\varphi) = M^\basis_{\basis^*}(a_\varphi)$, + \item $\Ker a_\varphi = V^\perp$, + \item $\varphi$ è non degenere se e solo se $M_\basis(\varphi)$ è invertibile, + \item $W^\perp = \Ker i^\top \circ a_\varphi$, + \item $a_\varphi(W^\perp) = \Ann(W) \cap \Imm a_\varphi$, + \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 $(W^\perp)^\perp = W^\dperp = W + \Rad(\varphi) = W + V^\perp$, + \item $(U + W)^\perp = U^\perp \cap W^\perp$, + \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 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). + \end{itemize} + + Se $\basis'$ è un'altra base di $V$, vale il seguente \textit{teorema di cambiamento di base}: + + \[ M_{\basis'}(\varphi) = M_{\basis}^{\basis'}(\Idv)^\top \, M_\basis(\varphi) \, M_{\basis}^{\basis'}(\Idv). \] + + Si definisce relazione di congruenza la relazione di equivalenza $\cong$ (o $\equiv$) definita + su $\Sym(n, \KK)$ nel seguente modo: + + \[ A \cong B \iff \exists P \in \GL(n, \KK) \mid A = P^\top B P. \] + + + \begin{itemize} + \item $A \cong B \implies \rg(A) = \rg(B)$ (il rango è invariante per congruenza; e dunque si può + definire $\rg(\varphi)$ come il rango di una qualsiasi matrice associata a $\varphi$), + \item $A \cong B \implies \det(A) \det(B) \geq 0$ (in $\KK = \RR$ il segno del determinante è invariante per congruenza), + \item Due matrici associate a $\varphi$ in basi diverse sono congruenti per la formula + di cambiamento di base. + \end{itemize} + + Si definiscono i seguenti tre indici per $\KK = \RR$: + + \begin{itemize} + \item $\iota_+ = \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} > 0 \}$, + \item $\iota_- = \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} < 0 \}$, + \item $\iota_0 = \dim V^\perp$, + \end{itemize} + + e si definisce segnatura di $\varphi$ la terna $\sigma = (\iota_+, \iota_-, \iota_0)$. + + Si dice che una base $\basis$ di $V$ è ortogonale se i suoi vettori sono a due a due ortogonali (e + quindi la matrice associata in tale base è diagonale). Se $\Char \KK \neq 2$, valgono i seguenti risultati: + + \begin{itemize} + \item $\varphi(\v, \w) = \frac{q(\v + \w) - q(\v) - q(\w)}{2}$ (formula di polarizzazione; $\varphi$ è + completamente determinata dalla sua forma quadratica), + + \item Esiste sempre una base ortogonale $\basis$ di $V$ (teorema di Lagrange; è sufficiente considerare + l'esistenza di un vettore anisotropo $\w \in V$ ed osservare che $V = W \oplus^\perp W^\perp$, dove $W = \Span(V)$, concludendo per induzione; o in caso di non esistenza di tale $\w$, concludere per il + risultato precedente), + + \item (se $\KK = \CC$) Esiste sempre una base ortogonale $\basis$ di $V$ tale che: + + \[ M_\basis(\varphi) = \Matrix{I_r & \rvline & 0 \\ \hline 0 & \rvline & 0\,}, \] + + \vskip 0.05in + + dove $r = \rg(\varphi)$ (teorema di Sylvester, caso complesso; si consideri una base ortogonale e se + ne normalizzino i vettori anisotropi), + + \item Due matrici simmetriche con stesso rango allora non solo sono SD-equivalenti, ma sono + anche congruenti, + + \item (se $\KK = \RR$) Esiste sempre una base ortogonale $\basis$ di $V$ tale che: + + \[ M_\basis(\varphi) = \Matrix{I_{\iota_+} & \rvline & 0 & \rvline & 0 \\ \hline 0 & \rvline & -I_{\iota_-} & \rvline & 0 \\ \hline 0 & \rvline & 0 & \rvline & 0\cdot I_{\iota_0} }. \] + + \vskip 0.05in + + Inoltre $\sigma$ è un invariante completo per la congruenza, e vale che, su una qualsiasi base ortogonale $\basis'$ di $V$, $\iota_+$ è esattamente il numero + di vettori anisotropi di base con forma quadratica positiva, che $\iota_-$ è il numero di vettori con forma + negativa e che $\iota_0$ è il numero di vettori isotropi (teorema di Sylvester, caso reale; si consideri + 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 $\varphi \leq 0 \iff \sigma = (0, n - k, k)$, + con $0 \leq k \leq n$ tale che $k = \dim V^\perp$, + + \item I vettori isotropi di una base ortogonale sono una base di $V^\perp$, + + \item $\rg(\varphi) = \iota_+ + \iota_-$, + + \item $n = \iota_+ + \iota_- + \iota_0$, + + \item Se $W$ è un sottospazio di $V$, $\iota_+(\varphi) \geq \iota_+(\restr{\varphi}{W})$ e + $\iota_-(\varphi) \geq \iota_-(\restr{\varphi}{W})$, + + \item Se $V = U \oplus^\perp W$, $\sigma(\varphi) = \sigma(\restr{\varphi}{U}) + \sigma(\restr{\varphi}{W})$, + + \item Se $\KK = \RR$ e $A = M_\basis(\varphi)$, allora: + \[ \sigma = \textstyle \left( \sum_{\substack{\lambda \in \Sp(\varphi) \\ \lambda > 0}} \mu_a(\lambda), \; \sum_{\substack{\lambda \in \Sp(A) \\ \lambda < 0}} \mu_a(\lambda), \; \mu_0(\lambda) \right), \] + come conseguenza del teorema spettrale reale. + \end{itemize} + + Si chiama matrice di Sylvester una matrice della forma vista nell'enunciato del teorema di Sylvester + reale, e si dice che una base $\basis$ è una base di Sylvester se la matrice ad essa associata è di + Sylvester. Per il teorema di Sylvester, tale base esiste sempre, e la matrice di Sylvester è unica per + ogni prodotto scalare $\varphi$. + + \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 + $\basis$ è anisotropo o al più vi è un vettore isotropo, posto in fondo come $\vv n$), si può + trovare una base ortogonale $\basis' = \{ \vv 1', \ldots, \vv n' \}$ a partire da $\basis$ tale che ne mantenga la stessa bandiera, ossia tale che: + + \[ \Span(\vv 1', \ldots, \vv i') = \Span(\vv 1, \ldots, \vv i) \forall 1 \leq i \leq n. \] + + Si definisce $C(\w, \v) = \frac{\varphi(\v, \w)}{\varphi(\w, \w)}$ come il coefficiente di Fourier + di $\v$ rispetto a $\w$. L'algoritmo allora funziona nel seguente modo: + + \begin{enumerate} + \item Si prenda in considerazione $\vv 1$ e si sottragga ad ogni altro vettore $\vv i$ della base il + vettore $C(\vv 1, \vv i) \, \vv 1$, + \item Si ripeta il processo considerando come $\basis$ tutti i vettori di $\basis$ con $\vv 1$ escluso, + o si termini l'algoritmo una volta che è rimasto un solo vettore. + \end{enumerate} + \subsubsection{Metodo di Jacobi per il calcolo della segnatura} + + Sia $A = M_\basis(\varphi)$ una matrice associata a $\varphi$ nella base $\basis$. + Sia $d_0 := 1$. Se $d_i = \det(A_{1, \ldots, i}^{1, \ldots, i})$ (è possibile anche + prendere un'altra sequenza di minori, a patto che essi siano principali e che siano + crescenti per inclusione) è diverso da zero + per ogni $1 \leq i \leq n-1$, allora $\iota_+$ è il numero di permanenze di segno + di $d_i$ (zero escluso), $\iota_-$ è il numero di variazioni di segno (zero escluso), e $\iota_0$ è $1$ se + $d_n = 0$ o $0$ altrimenti. + + In generale, se $W$ è un sottospazio di $W'$, $W$ ha codimensione $1$ rispetto a $W'$ e $\det(M_{\basis_W}(\restr{\varphi}{W})) \neq 0$ per una base $\basis_W$ di $W$, allora la segnatura + di $\restr{\varphi}{W'}$ è la stessa di $\restr{\varphi}{W}$, dove si aggiunge + $1$ a $\iota_+$, se i determinanti $\det(M_{\basis_W}(\restr{\varphi}{W}))$ e $\det(M_{\basis_{W'}}(\restr{\varphi}{W}))$ (dove $\basis_{W'}$ è una base di $W'$) concordano di segno, $1$ a $\iota_-$, se + sono discordi, o $1$ a $\iota_0$ se l'ultimo di questi due determinanti è nullo. + + Dal metodo di Jacobi si deduce il criterio di definitezza di Sylvester: $A$ è + definita positiva se e solo se $d_i > 0$ $\forall 1 \leq i \leq n$; $A$ è + definita negativa se e solo se $(-1)^i d_i > 0$ $\forall 1 \leq i \leq n$. + + \subsubsection{Sottospazi isotropi e indice di Witt} + + Si dice che un sottospazio $W$ di $V$ è isotropo se $\restr{\varphi}{W} = 0$, o + equivalentemente se $W \subseteq W^\perp$ (i.e.~se $W \cap W^\perp = W$, e quindi + se $\Rad(\restr{\varphi}{W}) = W$). Si definisce allora l'indice di Witt $W(\varphi)$ come + la dimensione massima di un sottospazio isotropo di $V$. + + \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 $\KK = \RR$, 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 $\varphi$ 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). + \end{itemize} \vfill \hrule diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index b6f7a4a..2b1db33 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -109,6 +109,8 @@ \newcommand{\conj}[1]{\overline{#1}} +\DeclareMathOperator{\PS}{PS} + \let\imm\Im \let\Im\undefined \DeclareMathOperator{\Im}{Im}