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+/Root 383 0 R +/Info 384 0 R +/ID [<211A11E7758D3E2B88709C111F63DACE> <211A11E7758D3E2B88709C111F63DACE>] >> startxref -411433 +417549 %%EOF diff --git a/Geometria 1/Scheda riassuntiva/main.tex b/Geometria 1/Scheda riassuntiva/main.tex index 1db6140..4294442 100644 --- a/Geometria 1/Scheda riassuntiva/main.tex +++ b/Geometria 1/Scheda riassuntiva/main.tex @@ -2016,6 +2016,48 @@ rango, allora $V$ e $W$ sono isometrici tra loro (come conseguenza stavolta del teorema di Sylvester complesso). + \subsection{Operatori simmetrici, ortogonali, hermitiani e unitari} + + [TODO] + + \subsection{Spazi euclidei reali e complessi} + + Si dice che $(V, \varphi)$ è uno spazio euclideo reale se $V$ è un $\RR$-spazio e se + $\varphi$ è un prodotto scalare definito positivo. Si dice che $(V_\CC, \varphi_\CC)$ è uno spazio euclideo complesso se $V_\CC$ è un $\CC$-spazio e se $\varphi_\CC$ è un + prodotto hermitiano definito positivo. + + Questi due tipi di spazi hanno in comune alcune proprietà particolari. Si definisce + innanzitutto la norma euclidea per uno spazio euclideo $(V, \varphi)$ come: + + \[ \norm{\v} = \sqrt{q(\v)} = \sqrt{\varphi(\v, \v)}. \] + + Tale norma soddisfa alcune proprietà: + + \begin{itemize} + \item $\norm{\lambda \v} = \abs{\lambda} \norm{\v}$, + \item $\norm{\v} \norm{\w} \geq \abs{\varphi(\v, \w)}$ (disuguaglianza di Cauchy-Schwarz), + \item $\norm{\v + \w} \leq \norm{\v} + \norm{\w}$ (disuguaglianza triangolare). + \end{itemize} + + Su questi due spazi possono essere definiti due particolare operatori: la + proiezione ortogonale e l'inversione ortogonale. + + Si definisce proiezione ortogonale su un sottospazio $W \neq \zerovecset$ l'operatore $\pr_W \in \End(V)$ tale + che $\pr_W(\v) = \w$, dove $\v = \w + \w^\perp$, con $\w \in W$ e $\w^\perp \in W^\perp$. Tale decomposizione è ben definita e unica dacché $V = W \oplus^\perp W^\perp$ (infatti $\varphi$ è definita positiva). Una proiezione ortogonale + soddisfa la relazione $\pr_W^2 = \pr_W$, da cui si ricava che $\varphi_{\pr_W} \mid x(x-1)$ (implicandone la diagonalizzabilità). Infatti $V_1 = \Ker(\pr_W - \Idv) = W$ e $V_0 = \Ker(\pr_W) = W^\perp$ (per cui $\varphi_{\pr_W}(x) = x(x-1)$). La + proiezione ortogonale è un operatore simmetrico (se lo spazio è euclideo reale) + o hermitiano (se lo spazio è euclideo complesso); infatti vale che + $\varphi(\pr_W(\v), \w) = \varphi(\pr_W(\v), \pr_W(\w) + \pr_{W^\perp}(\w)) = + \varphi(\pr_W(\v), \pr_W(\w)) = \varphi(\pr_W(\v) + \pr_{W^\perp}(\v), \pr_W(\w)) = + \varphi(\v, \pr_W(\w))$. + + Si definisce inversione ortogonale su un sottospazio $W \neq \zerovecset$ l'operatore $\rho_W \in \End(V)$ tale + che $\rho_W(\v) = \w - \w^\perp$, dove $\v = \w + \w^\perp$, con $\w \in W$ e $\w^\perp \in W^\perp$. Come prima, tale decomposizione è unica e ben definita. Un'inversione ortogonale + soddisfa la relazione $\rho_W^2 = \Idv$, da cui si ricava che $\varphi_{\rho_W} \mid (x+1)(x-1)$ (implicandone la diagonalizzabilità). Infatti $V_1 = \Ker(\rho_W - \Idv) = W$ e $V_{-1} = \Ker(\rho_W + \Idv) = W^\perp$ (per cui $\varphi_{\rho_W}(x) = (x+1)(x-1)$). Se $\dim W = \dim V - 1$, allora si dice che l'inversione ortogonale + è una riflessione ortogonale. L'inversione ortogonale è sempre un operatore + ortogonale (se lo spazio è euclideo reale) o unitario (se lo spazio è euclideo + complesso); infatti vale che $\varphi(\v, \w) = \varphi(\pr_W(\v) + \pr_{W^\perp}(\v), \pr_W(\w) + \pr_{W^\perp}(\w)) = \varphi(\pr_W(\v), \pr_W(\w)) + \varphi(\pr_{W^\perp}(\v), \pr_{W^\perp}(\w)) = \varphi(\pr_W(\v), \pr_W(\w)) + \varphi(-\pr_{W^\perp}(\v), -\pr_{W^\perp}(\w)) = \varphi(\pr_W(\v) - \pr_{W^\perp}(\v), \pr_W(\w) - \pr_{W^\perp}(\w)) = \varphi(\rho_W(\v), \rho_W(\w))$. + \subsection{Teorema spettrale reale e complesso} Se $f$ è simmetrico o hermitiano, esiste sempre una base ortonormale di autovettori