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00000 n +0000473034 00000 n +0000473072 00000 n +0000473200 00000 n trailer << /Size 449 /Root 447 0 R /Info 448 0 R -/ID [<6EA3B25F91DF9846EA90A42DA92EE5EC> <6EA3B25F91DF9846EA90A42DA92EE5EC>] >> +/ID [<77A46A226E6BCB15B4D84815CCD585DD> <77A46A226E6BCB15B4D84815CCD585DD>] >> startxref -473264 +473474 %%EOF diff --git a/Primo anno/Geometria 1/Scheda riassuntiva/main.tex b/Primo anno/Geometria 1/Scheda riassuntiva/main.tex index cb9c5b2..454bfb6 100644 --- a/Primo anno/Geometria 1/Scheda riassuntiva/main.tex +++ b/Primo anno/Geometria 1/Scheda riassuntiva/main.tex @@ -2000,6 +2000,11 @@ \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} + + Dal momento che l'algoritmo mantiene invariata la bandiera della base, + una matrice triangolabile è anche ortogonalmente triangolabile se la base + non contiene alcun (o contiene al più un) vettore isotropo secondo un + certo prodotto scalare. \subsubsection{Metodo di Jacobi per il calcolo della segnatura} Sia $A = M_\basis(\varphi)$ una matrice associata a $\varphi$ nella base $\basis$. @@ -2179,7 +2184,7 @@ 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 + ortogonale (se lo spazio euclideo è reale) o unitario (se lo spazio euclideo è 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} @@ -2205,7 +2210,11 @@ In termini matriciali, se $A$ è una matrice simmetrica a elementi reali (o hermitiana a elementi complessi), esiste una matrice $O \in O(n)$ (o $U \in U(n)$) tale per cui $O^\top A O$ (o $U \in U(n)$) è diagonale. Infatti $f_A$, l'operatore indotto da $A$ nella base ortonormale di $\RR^n$ (o $\CC^n$), è un operatore simmetrico (o hermitiano) rispetto al prodotto standard dello - spazio euclideo che si sta studiando. + spazio euclideo che si sta studiando. \\ + + Una matrice reale è simmetrica se e solo se è ortogonalmente diagonalizzabile. + Una matrice complessa è hermitiana se e solo se è unitariamente diagonalizzabile + con autovalori reali. \subsubsection{Radice quadrata di una matrice simmetrica, decomposizione polare e simultanea ortogonalizzabilità}