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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 4a088a9..eaddf09 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 @@ -626,11 +626,78 @@ \li Se $\v = a_1 \vv 1 + \ldots + a_n \vv n$, con $a_1$, ..., $a_n \in \KK$, si osserva che $\varphi(\v, \vv i) = a_i \varphi(\vv i, \vv i)$. Quindi $\v = \sum_{i=1}^n \frac{\varphi(\v, \vv i)}{\varphi(\vv i, \vv i)} \, \vv i$. In particolare, $\frac{\varphi(\v, \vv i)}{\varphi(\vv i, \vv i)}$ è - detto \textbf{coefficiente di Fourier} di $\v$ rispetto a $\vv i$. Se $\basis$ è ortonormale, + detto \textbf{coefficiente di Fourier} di $\v$ rispetto a $\vv i$, e si indica con $C(\v, \vv i)$. Se $\basis$ è ortonormale, $\v = \sum_{i=1}^n \varphi(\v, \vv i) \, \vv i$. \\ - \li Quindi $\norm{\v}^2 = \varphi(\v, \v) = \sum_{i=1}^n \frac{\varphi(\v, \vv i)^2}{\varphi(\vv i, \vv i)}$. In particolare, se $\basis$ è ortonormale, $\norm{\v}^2 = \sum_{i=1}^n \varphi(\v, \vv i)^2$. In tal caso, si può esprimere la disuguaglianza di Bessel: $\norm{\v}^2 \geq \sum_{i=1}^k \varphi(\v, \vv i)^2$ per $k \leq n$. \end{remark} + + \begin{remark} (algoritmo di ortogonalizzazione di Gram-Schmidt) + Se $\varphi$ è non degenere (o in generale, se $\CI(\varphi) = \zerovecset$) ed è + data una base $\basis = \{ \vv 1, \ldots, \vv n \}$ per $V$ (dove si ricorda che deve valere + $\Char \KK \neq 2$), è possibile + applicare l'\textbf{algoritmo di ortogonalizzazione di Gram-Schmidt} per ottenere + da $\basis$ una nuova base $\basis' = \{ \vv 1', \ldots, \vv n' \}$ con le seguenti proprietà: + + \begin{enumerate}[(i)] + \item $\basis'$ è una base ortogonale, + \item $\basis'$ mantiene la stessa bandiera di $\basis$ (ossia $\Span(\vv 1, \ldots, \vv i) = \Span(\vv 1', \ldots, \vv i')$ per ogni $1 \leq i \leq n$). + \end{enumerate} + + L'algoritmo si applica nel seguente modo: si prenda in considerazione $\vv 1$ e sottragga ad ogni altro vettore + della base il vettore $C(\vv 1, \vv i) \vv 1 = \frac{\varphi(\vv 1, \vv i)}{\varphi(\vv 1, \vv 1)} \vv 1$, + rendendo ortogonale ogni altro vettore della base con $\vv 1$. Pertanto si applica la mappa + $\vv i \mapsto \vv i - \frac{\varphi(\vv 1, \vv i)}{\varphi(\vv 1, \vv 1)} \vv i = \vv i ^{(1)}$. + Si verifica infatti che $\vv 1$ e $\vv i ^{(1)}$ sono ortogonali per $2 \leq i \leq n$: + + \[ \varphi(\vv 1, \vv i^{(1)}) = \varphi(\vv 1, \vv i) - \varphi\left(\vv 1, \frac{\varphi(\vv 1, \vv i)}{\varphi(\vv 1, \vv 1)} \vv i\right) = \varphi(\vv 1, \vv i) - \varphi(\vv 1, \vv i) = 0. \] + + Poiché $\vv 1$ non è isotropo, si deduce la decomposizione $V = \Span(\vv 1) \oplus \Span(\vv 1)^\perp$. + In particolare $\dim \Span(\vv 1)^\perp = n-1$: essendo allora i vettori $\vv 2 ^{(1)}, \ldots, \vv n ^{(1)}$ + linearmente indipendenti e appartenenti a $\Span(\vv 1)^\perp$, ne sono una base. Si conclude quindi + che vale la seguente decomposizione: + + \[ V = \Span(\vv 1) \oplus^\perp \Span(\vv 2 ^{(1)}, \ldots, \vv n ^{(1)}). \] + + \vskip 0.05in + + Si riapplica dunque l'algoritmo di Gram-Schmidt prendendo come spazio vettoriale lo spazio generato dai + vettori a cui si è applicato precedentemente l'algoritmo, ossia $V' = \Span(\vv 2 ^{(1)}, \ldots, \vv n ^{(1)})$, + fino a che non si ottiene $V' = \zerovecset$. \\ + + Si può addirittura ottenere una base ortonormale a partire da $\basis'$ normalizzando ogni vettore (ossia + dividendo per la propria norma), se si sta considerando uno spazio euclideo. + \end{remark} + + \begin{remark} + Poiché la base ottenuta tramite Gram-Schmidt mantiene la stessa bandiera della base di partenza, + ogni matrice triangolabile è anche triangolabile mediante una base ortogonale. + \end{remark} + + \begin{example} + Si consideri $V = (\RR^3, \innprod{\cdot})$, ossia di $\RR^3$ dotato del prodotto scalare standard, e + si applichi l'algoritmo di ortogonalizzazione di Gram-Schmidt sulla seguente base: + + \[ \basis = \Biggl\{ \underbrace{\Vector{1 \\ 0 \\ 0}}_{\vv 1 \, = \, \e1}, \underbrace{\Vector{1 \\ 1 \\ 0}}_{\vv 2}, \underbrace{\Vector{1 \\ 1 \\ 1}}_{\vv 3} \Biggl\} \] + + \vskip 0.05in + + Alla prima iterazione dell'algoritmo si ottengono i seguenti vettori: + + \begin{itemize} + \item $\vv 2 ^{(1)} = \vv 2 - \frac{\varphi(\vv 1, \vv 2)}{\varphi(\vv 1, \vv 1)} \vv 1 = \vv 2 - \vv 1 = \Vector{0 \\ 1 \\ 0} = \e 2$, + \item $\vv 3 ^{(1)} = \vv 3 - \frac{\varphi(\vv 1, \vv 3)}{\varphi(\vv 1, \vv 1)} \vv 1 = \vv 3 - \vv 1 = \Vector{0 \\ 1 \\ 1}$. + \end{itemize} + + Si considera ora $V' = \Span(\vv 2 ^{(1)}, \vv 3 ^{(1)})$. Alla seconda iterazione dell'algoritmo si + ottiene allora il seguente vettore: + + \begin{itemize} + \item $\vv 3 ^{(2)} = \vv 3 ^{(1)} - \frac{\varphi(\vv 2 ^{(1)}, \vv 3 ^{(1)})}{\varphi(\vv 2 ^{(1)}, \vv 2 ^{(1)})} \vv 2 ^{(1)} = \vv 3 ^{(1)} - \vv 2 ^{(1)} = \Vector{0 \\ 0 \\ 1} = \e 3$. + \end{itemize} + + Quindi la base ottenuta è $\basis' = \{\e1, \e2, \e3\}$, ossia la base canonica di $\RR^3$, già + ortonormale. + \end{example} \end{document} \ No newline at end of file