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deve allora per forza valere $\varphi(\v, \w) = 0$. - % TODO: sistemare - Se $f$ è simmetrico o hermitiano, esiste sempre una base ortonormale di autovettori - per $f$. $f$ è normale se e solo se è diagonalizzabile. Esiste sempre $S \in \Sym(n, \RR)$ tale per cui $S^2 = A$, con $A \in \Sym(n, \RR)$. Se $S$ è definita positiva, - tale $S$ è unica. Se $A \in \GL(n, \RR)$, esistono e sono unici $P \in O(n)$, + Se $f$ è simmetrico o hermitiano, esiste sempre una + base ortonormale di autovettori (\textit{teorema spettrale}). Se così non fosse, detto $W = V_{\lambda_1} \oplus^\perp \cdots \oplus^\perp V_{\lambda_k}$, $W^\perp$ sarebbe $f$-invariante e simmetrico/hermitiano, e dunque ammetterebbe un autovalore reale, contrariamente a quanto ipotizzato, \Lightning. Alternativamente, poiché + $f$ è simmetrico (e in tal caso anche perché il + polinomio caratteristico è completamente fattorizzabile in $\RR$) o hermitiano, $f$ è anche + normale, ed è dunque diagonalizzabile; allora, poiché gli autospazi sono in somma diretta ortogonale, $f$ è anche ortogonalmente o unitariamente diagonalizzabile. + + \subsubsection{Radice quadrata di una matrice simmetrica, decomposizione polare e simultanea ortogonalizzabilità} + + Se $A \in \Sym(n, \RR)$ è semidefinita positiva, allora + esiste sempre una matrice $S \in \Sym(n, \RR)$ tale + per cui $S^2 = A$. Se si suppone anche che $S$ è + semidefinita positiva, tale matrice diventa unica e + viene detta \textit{radice quadrata} di $A$, indicata come $\sqrt{A}$. \\ + + Per costruire tale radice quadrata è sufficiente + considerare $P \in O(n)$ tale per cui + $P^\top A P = D$, dove $D \in M(n, \RR)$ è + diagonale, secondo il teorema spettrale. Poiché $A$ è semidefinita positiva, $D$ + si compone di soli elementi non negativi, ed è + dunque possibile costruire la matrice $\sqrt{D} \in M(n, \RR)$ dove $\sqrt{D}_{ii} = \sqrt{D_{ii}}$ (da cui si deduce che $\sqrt{D}^2 = D$ e che $\sqrt{D}$ è esattamente la radice quadrata di $D$). + Si consideri dunque $S = P \sqrt{D} P^\top$; vale + che $S^2 = P D P^\top = A$, e dunque $S$ è la + radice quadrata $\sqrt{A}$ di $A$ (per dimostrare l'unicità di tale matrice è sufficiente ridursi all'uguaglianza negli autospazi). Si osserva che + se $A$ è definita positiva, anche $S$ lo è. \\ \vskip 0.05in + + Se $A \in M(n, \RR)$ esistono e sono uniche le + matrici $P \in O(n)$, $S \in \Sym(n, \RR)$, con $S$ semidefinita positiva, tali per + cui $A = PS$. In particolare vale che $S = \sqrt{A A^\top}$; se dunque $A \in \GL(n, \RR)$, $S$ è + definita positiva (in tal caso $\Ker A^\top A = \Ker A = \zerovecset$ -- come visto nella sezione sulle matrici --, e dunque $\vec x^\top A^\top A \vec x = \innprod{A \vec x, A \vec x} > 0$ $\implies A^\top A > 0 \implies S > 0$). + + Se $A \in \GL(n, \RR)$, esistono e sono unici $P \in O(n)$, $S \in \Sym(n, \RR)$ tali per cui $A = PS$ (in particolare $S = \sqrt{A A^\top}$). - Se $\varphi$ è definito positivo e $\psi$ è un altro prodotto scalare, allora - i due prodotti sono simultaneamente ortogonalizzabili. È sufficiente prendere + Due prodotti $\varphi$, $\psi$ si dicono simultaneamente ortogonalizzabili se esiste una + base $\basis$ tale per cui sia che $M_\basis(\varphi)$ che $M_\basis(\psi)$ sono + diagonali (ossia se esiste una base ortogonale per + entrambi i prodotti). \\ \vskip 0.05in + + Se $\KK=\RR$ o $\KK=\CC$, $\varphi$ è definito positivo, allora + i due prodotti $\varphi$ e $\psi$ sono sempre simultaneamente ortogonalizzabili. È sufficiente infatti prendere una base $\basis$ ortonormale di $\varphi$, e trovare la base ortonormale $\basis'$ di autovettori che rende $M_\basis(\psi)$ diagonale. In tale base $\basis'$, $M_{\basis'}(\varphi)$ è l'identità e $M_\basis(\psi)$ è diagonale: dunque la base è ortogonale per ambo - i prodotti scalari. + i prodotti. \subsection{Azioni di gruppo} Sia $G$ un gruppo e $X$ un insieme. Un'azione sinistra\footnote{Un'azione sinistra induce sempre anche un'azione destra, ponendo $x \cdot g=g^{-1} \cdot x$.} di $G$ su $X$ a sinistra un'applicazione $\cdot : G \times X \rightarrow X$, per la quale si pone $g \cdot x := \cdot(g, x)$, tale che: