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zKV>w9U7U;9l%T<~<#B?^VrmAMI;<*3YfRN%Z3T0{qcnz)IOL!vGMG^;BTt&=z;x*C z&EUcauxY|r*t9$RzYaeF9c+OBBU|ZAXdP=H)tpTf7*;zuj_Bi2np*ss5kb2l40MwV zHe1$XR;o5KcrwWrD|03{nz+Q?g`aD*tj7}Yw>U;tH=_knbr?}H{SKSODHhQUne{{X zMUyi|QYzLv$;oyJOQ~!+f;9xoqBu(6XY2^#-@ta?)vlXb&^V#xv*p3@o5SU7dwc%y z#qkH+G8CK>pP!z#(;7Bx_pnb+j!y9oW~|@s&0lTjQg=b+r2BD!4D9aZJxu2PwAd|# l(~s`Vcm6N)dX8h@+p~k^+56?;(G2F69A{fwkDu+&{s9Cr#}5Di diff --git a/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex index 485976a..b08436e 100644 --- a/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex +++ b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.tex @@ -1053,4 +1053,127 @@ Quindi $\varphi(\rho_W(\vv 1), \rho_W(\vv 2)) = \varphi(\ww 1, \ww 2) + \varphi(\ww 1', \ww 2) + \varphi(\ww 1, \ww 2') + \varphi(\ww 1', \ww 2') = \varphi(\vv 1, \vv 2)$. \end{remark} + + \begin{theorem} + Ogni isometria è prodotto di al più $n+1$ riflessioni. + \end{theorem} + + \hr + + \begin{lemma} + Sia $f \in \End(V)$ simmetrico (o hermitiano). Allora $f$ ha solo autovalori reali\footnote{Nel caso + di $f$ simmetrico, si intende in particolare che tutte le radici del suo polinomio caratteristico + sono reali.}. + \end{lemma} + + \begin{proof} + Si assuma che $V$ è uno spazio euclideo complesso, e quindi che $\varphi$ è un prodotto hermitiano. Allora, + se $f$ è hermitiano, sia $\lambda \in \CC$ un suo autovalore\footnote{Tale autovalore esiste sicuramente dal momento + che $\KK = \CC$ è un campo algebricamente chiuso.} e sia $\v \in \lambda$. Allora $\varphi(\v, f(\v)) = + \varphi(f(\v), \v) = \conj{\varphi(\v, f(\v))} \implies \varphi(\v, f(\v)) \in \RR$. Inoltre vale + la seguente identità: + + \[ \varphi(\v, f(\v)) = \varphi(\v, \lambda \v) = \lambda \varphi(\v, \v), \] + + da cui, ricordando che $\varphi$ è non degenere e che $\varphi(\v, \v) \in \RR$, si ricava che: + + \[ \lambda = \frac{\varphi(\v, f(\v))}{\varphi(\v, \v)} \in \RR. \] + + \vskip 0.05in + + Sia ora invece $V$ è uno spazio euclideo reale e $\varphi$ è un prodotto scalare. Allora, $(V_\CC, \varphi_\CC)$ + è uno spazio euclideo complesso, e $f_\CC$ è hermitiano. Sia $\basis$ una base di $V$. Allora, come visto all'inizio di questa + dimostrazione, $f_\CC$ ha solo autovalori reali, da cui si ricava che il polinomio caratteristico + di $f_\CC$ è completamente riducibile in $\RR$. Si osserva inoltre che $p_f(\lambda) = \det(M_\basis(f) - \lambda I_n) = \det(M_\basis(f_\CC) - \lambda I_n) = p_{f_\CC}(\lambda)$. Si conclude dunque che + anche $p_f$ è completamente riducibile in $\RR$. + \end{proof} + + \begin{remark} + Dal lemma precedente consegue immediatamente che se $A \in M(n, \RR)$ è simmetrica (o se appartiene a + $M(n, \CC)$ ed è hermitiana), considerando l'operatore simmetrico $f_A$ indotto da $A$ in $\RR^n$ (o $\CC^n$), + $f_A$ ha tutti autovalori reali, e dunque così anche $A$. + \end{remark} + + \begin{lemma} + Sia $f \in \End(V)$ simmetrico (o hermitiano). Allora se $\lambda$, $\mu$ sono due autovalori distinti + di $f$, $V_\lambda \perp V_\mu$. + \end{lemma} + + \begin{proof} + Siano $\v \in V_\lambda$ e $\w \in V_\mu$. Allora\footnote{Si osserva che non è stato coniugato $\lambda$ + nei passaggi algebrici, valendo $\lambda \in \RR$ dallo scorso lemma.} $\lambda \varphi(\v, \w) = \varphi(\lambda \v, \w) = \varphi(f(\v), \w) = \varphi(\v, f(\w)) = \varphi(\v, \mu \w) = \mu \varphi(\v, \w)$. + Pertanto vale la seguente identità: + + \[ (\lambda - \mu) \varphi(\v, \w) = 0. \] + + \vskip 0.05in + + In particolare, valendo $\lambda - \mu \neq 0$ per ipotesi, $\varphi(\v, \w) = 0 \implies V_\lambda \perp V_\mu$, + da cui la tesi. + \end{proof} + + \begin{lemma} + Sia $f \in \End(V)$ simmetrico (o hermitiano). Se $W \subseteq V$ è $f$-invariante, allora anche + $W^\perp$ lo è. + \end{lemma} + + \begin{proof} + Siano $\w \in W$ e $\v \in W^\perp$. Allora $\varphi(\w, f(\v)) = \varphi(\underbrace{f(\w)}_{\in \, W}, \v) = 0$, da cui si ricava che $f(\v) \in W^\perp$, ossia la tesi. + \end{proof} + + \begin{theorem} [spettrale reale] + Sia $(V, \varphi)$ uno spazio euclideo reale (o complesso) e sia $f \in \End(V)$ simmetrico (o hermitiano). Allora esiste una base ortogonale $\basis$ di $V$ composta di autovettori per $f$. + \end{theorem} + + \begin{proof} + Siano $\lambda_1$, ..., $\lambda_k$ tutti gli autovalori reali di $f$. Sia inoltre + $W = V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k}$. Per i lemmi precedenti, + vale che: + + \[ W = V_{\lambda_1} \oplus^\perp \cdots \oplus^\perp V_{\lambda_k}. \] + + \vskip 0.05in + + Sicuramente $W \subset V$. Si assuma però che $W \subsetneq V$. Allora $V = W \oplus^\perp W^\perp$. In particolare, per il lemma + precedente, $W^\perp$ è $f$-invariante. Quindi $\restr{f}{W^\perp}$ è un endomorfismo + di uno spazio di dimensione non nulla. Si osserva che $\restr{f}{W^\perp}$ è chiaramente + simmetrico (o hermitiano), essendo solo una restrizione di $f$. Allora $\restr{f}{W^\perp}$ ammette + autovalori reali per i lemmi precedenti; tuttavia questo è un assurdo, dal momento che ogni autovalore di $\restr{f}{W^\perp}$ è anche autovalore di $f$ e si era supposto che\footnote{Infatti tale autovalore $\lambda$ + non può già comparire tra questi autovalori, altrimenti, detto $i \in \NN$ tale che $\lambda = \lambda_i$, $V_{\lambda_i} \cap W^\perp \neq \zerovecset$, violando la somma diretta supposta.} $\lambda_1$, ..., $\lambda_k$ fossero + tutti gli autovalori di $f$, \Lightning. Quindi $W = V$. Pertanto, detta $\basis_i$ una base ortonormale + di $V_{\lambda_i}$, $\basis = \cup_{i=1}^k \basis_i$ è una base ortonormale di $V$, da cui la tesi. + \end{proof} + + \begin{corollary} [teorema spettrale per le matrici] + Sia $A \in M(n, \RR)$ simmetrica (o appartenente a $M(n, \CC)$ ed hermitiana). Allora + $\exists P \in O_n$ (o $P \in U_n$) tale che $P\inv A P = P^\top A P$ (o $P\inv A P = P^* A P$ nel caso hermitiano) + sia una matrice diagonale reale. + \end{corollary} + + \begin{proof} + Si consideri $f_A$, l'operatore indotto dalla matrice $A$ in $\RR^n$ (o $\CC^n$). Allora + $f_A$ è un operatore simmetrico (o hermitiano) sul prodotto scalare (o hermitiano) standard. + Pertanto, per il teorema spettrale reale, esiste una base ortonormale $\basis = \{ \vv 1, \ldots, \vv n\}$ composta di autovettori + di $f_A$. In particolare, detta $\basis'$ la base canonica di $\RR^n$ (o $\CC^n$), vale + la seguente identità: + + \[ M_\basis(f) = M_{\basis'}^{\basis}(\Id)\inv M_{\basis'}(f) M_{\basis'}^{\basis}(\Id), \] + + dove $M_{\basis'}(f) = A$, $M_\basis(f)$ è diagonale, essendo $\basis$ composta di autovettori, e $P = M_{\basis'}^{\basis}$ + si configura nel seguente modo: + + \[ M_{\basis'}^{\basis}(f) = \Matrix{ \vv 1 & \rvline & \cdots & \rvline & \vv n }. \] + + Dacché $\basis$ è ortogonale, $P$ è anch'essa ortogonale, da cui la tesi. + \end{proof} + + \begin{remark}\nl + \li Un importante risultato che consegue direttamente dal teorema spettrale per le matrici riguarda + la segnatura di un prodotto scalare (o hermitiano). Infatti, detta $A = M_\basis(\varphi)$, + $D = P^\top A P$, e dunque $D \cong A$. Allora, essendo $D$ diagonale, l'indice di positività + è esattamente il numero di valori positivi sulla diagonale, ossia il numero di autovalori + positivi di $A$. Analogamente l'indice di negatività è il numero di autovalori negativi, + e quello di nullità è la molteplicità algebrica di $0$ come autovalore (ossia esattamente + la dimensione di $V^\perp_\varphi = \Ker a_\varphi$). + \end{remark} \end{document} \ No newline at end of file