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z6nF%TXBdcR^uIy)zx?lbUc+Dg0HYWn@V^ZMtc=0W3E^O%>rNN}au+h*eO_#us?P2}P3%YC1U`KX?M1IeucQ=xF5xd;{A0&o>!X0}%)?F~DRn za?(&)88}7~frLp)$^dU?1O}lLOb#t4#`>=c>i^L(Vg(*Kxw)IVx_i4?T9E>-0U=Gw K#U-mINBV!bhdKfP 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 188b016..318c0fe 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 @@ -4,7 +4,7 @@ \title{\textbf{Note del corso di Geometria 1}} \author{Gabriel Antonio Videtta} -\date{17 e 19 aprile 2023} +\date{17, 19 e 26 aprile 2023} \begin{document} @@ -143,7 +143,9 @@ \li Vale il teorema di Lagrange, e quindi quello di Sylvester, benché con alcune accortezze: si introduce, come nel caso di $\RR$, il concetto di segnatura, che diventa l'invariante completo della nuova congruenza hermitiana, che ancora una volta si dimostra essere una relazione - di equivalenza. + di equivalenza. \\ + \li Come mostrato nei momenti finali del documento (vd.~\textit{Esercizio 3}), vale + la formula delle dimensioni anche nel caso del prodotto hermitiano. \end{remark} \hr @@ -1335,9 +1337,10 @@ $a_{ij} = 0$ \, $\forall i < j \leq n$, dimostrando il passo induttivo, e quindi la tesi. \end{proof} - \begin{remark} - Chiaramente vale anche il viceversa del precedente lemma: se infatti $A \in M(n, \CC)$ è diagonale, - $A$ è anche normale, dal momento che commuta con $A^*$. + \begin{remark}\nl + \li Chiaramente vale anche il viceversa del precedente lemma: se infatti $A \in M(n, \CC)$ è diagonale, + $A$ è anche normale, dal momento che commuta con $A^*$. \\ + \li Reiterando la stessa dimostrazione del precedente lemma per $A \in M(n, \RR)$ triangolare superiore e normale reale (i.e.~$AA^\top = A^\top A$) si può ottenere una tesi analoga. \end{remark} \begin{theorem} @@ -1383,40 +1386,150 @@ \end{proof} \begin{remark}\nl - \li Si può osservare mediante l'ultimo corollario che $A$ hermitiana - $\implies$ $U^* A U = D$ è reale. \\ - \li Si può estendere il teorema ad una matrice generica - $A \in M(n, \RR)$ normale, a patto che $A$ ha tutti autovalori - reali. Infatti, in tal caso, esiste $O \in O_n$ tale che - $O^\top A O = D$ con $D$ diagonale. Inoltre, $A^\top = (O D O^\top)^\top - = O D^\top O^\top = O D O^\top = A$. Quindi $A$ è necessariamente - anche simmetrica. + \li Si può osservare mediante l'applicazione dell'ultimo corollario che, se $A$ è hermitiana (ed è dunque + anche normale), + $\exists U \in U_n \mid U^* A U = D$, dove $D \in M(n, \RR)$, ossia tale + corollario implica il teorema spettrale in forma complessa. Infatti + $\conj{D} = D^* = U^* A^* U = U^* A U = D \implies D \in M(n, \RR)$. \\ + + \li Se $A \in M(n, \RR)$ è una matrice normale reale (i.e.~$A A^\top = A^\top A$) con + $p_A$ completamente riducibile in $\RR$, allora è possibile reiterare la dimostrazione + del precedente teorema per concludere che $\exists O \in O_n \mid O^\top A O = D$ con + $D \in M(n, \RR)$, ossia che $A = O D O^\top$. + Tuttavia questo implica che $A^\top = (O D O^\top) = O D^\top O^\top = O D O^\top = A$, + ossia che $A$ è simmetrica. In particolare, per il teorema spettrale reale, vale + anche il viceversa. Pertanto, se $A \in M(n, \RR)$, $A$ è una matrice normale reale con $p_A$ completamente + riducibile in $\RR$ $\iff$ $A = A^\top$. \end{remark} - - \begin{exercise}\nl + + \begin{exercise} + Sia $V$ uno spazio dotato del prodotto $\varphi$. Sia + $W \subseteq V$ un sottospazio di $V$. Sia $\basis_W= \{ \ww 1, \ldots, \ww k \}$ + una base di $W$ e sia $\basis = \{ \ww 1, ..., \ww k, \vv{k+1}, ..., \vv n \}$ una base di $V$. + Sia $A = M_\basis(\varphi)$. Si dimostrino allora i seguenti risultati. + \begin{enumerate}[(i)] - \item $(V, \varphi)$ con $\varphi$ non degenere. - $W \subseteq V$ sottospazio. Sia $\ww 1$, ..., $\ww k$ - una base di $W$ e sia $\basis = \{ \ww 1, ..., \ww k, ..., \vv n \}$ una base di $V$. Allora $W^\top = \{ \v \in V \mid \varphi(\v, \ww i) = 0 \} = \{ \v \in V \mid A_{1,\ldots,k} [\v]_\basis = 0 \}$ con $A = M_\basis(\varphi)$. Pertanto $\dim W^\top = n - \rg(A_{1,...,k})$. - - \item Sia $U \subseteq V$ sottospazio. Dimostrare che - nel quoziente $V/U$ il prodotto induce il prodotto $\tilde \varphi([\v], [\v']) = \varphi(\v, \v')$ se e soltanto se - $U \subseteq V^\perp$ ($U \perp V$). - - \item Dimostrare che il prodotto $\tilde \varphi$ è - non degenere. - - \item Sia $\pi : V \to V/V^\perp$ la proiezione al - quoziente. Sia $W \subseteq V$ sottospazio. Dimostrare - che $W^\perp = \{ \v \in V \mid \tilde \varphi(\pi(\v), \varphi(\w)) = 0 \forall \w \in W \} = \pi\inv(\pi(W)^\perp)$. + \item $W^\perp = \{ \v \in V \mid \varphi(\v, \ww i) = 0 \}$, + \item $W^\perp = \{ \v \in V \mid A_{1,\ldots,k} [\v]_\basis = 0 \} = [\cdot]_\basis\inv (\Ker A_{1,\ldots,k})$, + \item $\dim W^\top = \dim V - \rg(A_{1,\ldots,k})$, + \item Se $\varphi$ è non degenere, $\dim W + \dim W^\perp = \dim V$. + \end{enumerate} + \end{exercise} + + \begin{proof}[Soluzione] + Chiaramente vale l'inclusione $W^\perp \subseteq \{ \v \in V \mid \varphi(\v, \ww i) = 0 \}$. Sia + allora $\v \in V \mid \varphi(\v, \ww i) = 0$ $\forall 1 \leq i \leq k$ e sia $\w \in W$. Allora esistono $\alpha_1$, ..., $\alpha_k$ tali + che $\w = \alpha_1 \ww 1 + \ldots + \alpha_k \ww k$. Pertanto si conclude che $\varphi(\v, \alpha_1 \ww 1 + \ldots + \alpha_k \ww k) = \alpha_1 \varphi(\v, \ww 1) + \ldots + \alpha_k \varphi(\v, \ww k) = 0 \implies \v \in W^\top$. Pertanto $W^\top = \{ \v \in V \mid \varphi(\v, \ww i) = 0 \}$, dimostrando (i). \\ + + Si osserva che $\varphi(\v, \ww i) = 0 \iff \varphi(\ww i, \v) = 0$. Se $\varphi$ è scalare, allora + $\varphi(\ww i, \v) = 0 \defiff [\ww i]_\basis^\top A [\v]_\basis = (\e i)^\top A [\v]_\basis = A_i [\v]_\basis = 0$. Pertanto $\v \in W^\top \iff A_i [\v]_\basis = 0$ $\forall 1 \leq i \leq k$, ossia se + $A_{1, \ldots, k} [\v]_\basis = 0$ e $[\v]_\basis \in \Ker A_{1, \ldots, k}$, dimostrando (ii). Analogamente + si ottiene la tesi se $\varphi$ è hermitiano. + Applicando la formula delle dimensioni, si ricava dunque che $\dim W^\top = \dim \Ker A_{1, \ldots, k} = + \dim V - \rg A_{1, \ldots, k}$, dimostrando (iii). \\ + + Se $\varphi$ è non degenere, $A$ è invertibile, dacché $\dim V^\perp = \dim \Ker A = 0$. Allora + ogni minore di taglia $k$ di $A$ ha determinante diverso da zero. Dacché ogni minore di taglia $k$ + di $A_{1,\ldots,k}$ è anche un minore di taglia $k$ di $A$, si ricava che anche ogni minore di taglia + $k$ di $A_{1, \ldots, k}$ ha determinante diverso da zero, e quindi che $\rg(A_{1,\ldots,k}) \geq k$. + Dacché deve anche valere $\rg(A_{1,\ldots,k}) \leq \min\{k,n\} = k$, si conclude che $\rg(A_{1,\ldots,k})$ + vale esattamente $k = \dim W$. Allora, dal punto (iii), vale che $\dim W^\perp + \dim W = \dim W^\perp + \rg(A_{1,\ldots,k}) = \dim V$, dimostrando il punto (iv). + \end{proof} + + \begin{exercise} + Sia $V$ uno spazio dotato del prodotto $\varphi$. Sia + $U \subseteq V$ un sottospazio di $V$. Si dimostrino allora i seguenti due + risultati. + + \begin{enumerate}[(i)] + \item Il prodotto $\varphi$ + induce un prodotto $\tilde \varphi : V/U \times V/U \to \KK$ tale che + $\tilde \varphi(\v + U, \v' + U) = \varphi(\v, \v')$ se e soltanto se $U \subseteq V^\perp$, ossia + se e solo se $U \perp V$. - \item Dedurre dai precedenti punti la formula della dimensione - dell'ortogonale. + %TODO: controllare che debba valere $U = V^\perp$ + \item Se $U = V^\perp$, allora il prodotto $\tilde \varphi$ è non degenere. - \item Dimostrare che $(W^\perp)^\perp = W + V^\perp$. + \item Sia $\pi : V \to V/V^\perp$ l'applicazione lineare di proiezione al quoziente. Allora + $U^\perp = \{ \v \in V \mid \tilde \varphi(\pi(\v), \pi(\U)) = 0 \, \forall \U \in U \} = \pi\inv(\pi(U)^\perp)$. - \item Dimostrare che $A$ anti-hermitiana è normale e ha tutti autovalori - immaginari. + \item Vale la formula delle dimensioni per il prodotto $\varphi$: $\dim U + \dim U^\perp = \dim V + \dim (U \cap V^\perp)$. \end{enumerate} \end{exercise} + + \begin{proof}[Soluzione] + Sia $\w = \v + \uu 1 \in \v + U$, con $\uu 1 \in U$. + Se $\tilde \varphi$ è ben definito, allora deve valere l'uguaglianza $\varphi(\v, \v') = \varphi(\w, \v') = + \varphi(\v + \uu 1, \v') = \varphi(\v, \v') + \varphi(\uu 1, \v')$, ossia $\varphi(\uu 1, \v') = 0$ $\forall \v' \in V \implies \uu 1 \in V^\perp \implies U \subseteq V^\perp$. Viceversa, se $U \subseteq V^\perp$, + sia $\w' = \v' + \uu 2 \in \v' + U$, con $\uu 2 \in U$. Allora vale la seguente identità: + + \[ \varphi(\w, \w') = \varphi(\v + \uu 1, \v' + \uu 2) = \varphi(\v, \v') + \underbrace{\varphi(\v, \uu 2) + \varphi(\uu 1, \v') + \varphi(\uu 1, \uu 2)}_{=\,0}. \] + + Pertanto $\tilde \varphi$ è ben definito, dimostrando (i). \\ + + Sia ora $U = V/V^\perp$. Sia $\v + U \in (V/U)^\perp = \Rad(\tilde \varphi)$. Allora, $\forall \v' + U \in V/U$, + $\tilde \varphi(\v + U, \v' + U) = \varphi(\v, \v') = 0$, ossia $\v \in V^\perp = U$. Pertanto + $\v + U = U \implies \Rad(\tilde \varphi) = \{ V^\perp \}$, e quindi $\tilde \varphi$ è non degenere, + dimostrando (ii). \\ + + Si dimostra adesso l'uguaglianza $U^\perp = \pi\inv(\pi(U)^\perp)$. Sia $\v \in U^\perp$. Allora + $\tilde \varphi(\pi(\v), \pi(\U)) = \tilde \varphi(\v + V^\perp, \U + V^\perp) = \varphi(\v, \U) = 0$ $\forall + \U \in U$, da cui si ricava che vale l'inclusione $U^\perp \subseteq \pi\inv(\pi(U)^\perp)$. Sia + ora $\v \in \pi\inv(\pi(U)^\perp)$, e sia $\U \in U$. Allora $\varphi(\v, \U) = \tilde \varphi(\v + V^\perp, \U + V^\perp) = \tilde \varphi(\pi(\v), \pi(\U)) = 0$, da cui vale la doppia inclusione, e dunque l'uguaglianza + desiderata, dimostrando (iii). \\ + + Dall'uguaglianza del punto (iii), l'applicazione della formula delle dimensioni e l'identità + ottenuta dal punto (iv) dell'\textit{Esercizio 2} rispetto al prodotto $\tilde \varphi$ non degenere, si ricavano + le seguenti identità: + + \[ \system{ \dim \pi(U) = \dim U - \dim (U \cap \Ker \pi) = \dim U - \dim (U \cap V^\perp), \\ \dim \pi(U)^\perp = \dim V/V^\perp - \dim \pi(U) = \dim V - \dim V^\perp - \dim \pi(U), \\ \dim U^\perp = \dim \pi(U)^\perp + \dim \Ker \pi = \dim \pi(U)^\perp + \dim V^\perp, } \] + + dalle quali si ricava la seguente identità: + + \[ \dim U^\perp = \dim V - \dim V^\perp - (\dim U - \dim(U \cap V^\perp)) + \dim V^\perp, \] + + \vskip 0.05in + + da cui si ricava che $\dim U + \dim U^\perp = \dim V + \dim(U \cap V^\perp)$, dimostrando (iv). + \end{proof} + + \begin{exercise} Sia $V$ uno spazio vettoriale dotato del prodotto $\varphi$. Si dimostri allora che $(W^\perp)^\perp = W + V^\perp$. + \end{exercise} + + \begin{proof}[Soluzione] + Sia $\v = \w' + \v' \in W + V^\perp$, con $\w' \in W$ e $\v' \in V^\perp$. Sia inoltre $\w \in W^\perp$. + Allora $\varphi(\v, \w) = \varphi(\w' + \v', \w) = \varphi(\w', \w) + \varphi(\v', \w) = 0$, + dove si è usato che $\w' \perp \w$ dacché $\w \in W^\perp$ e $\w' \in W$ e che $\v' \in V^\perp$. Allora + vale l'inclusione $W + V^\perp \subseteq (W^\perp)^\perp$. \\ + + Applicando le rispettive formule delle dimensioni a $W^\perp$, $(W^\perp)^\perp$ e $W + V^\perp$ si ottengono le seguenti identità: + + \[ \system{ \dim W^\perp = \dim V + \dim (W \cap V^\perp) - \dim W, \\ \dim (W^\perp)^\perp = \dim V + \dim (W^\perp \cap V^\perp) - \dim W^\perp, \\ \dim (W + V^\perp) = \dim W + \dim V^\perp - \dim (W \cap V^\perp), } \] + + \vskip 0.05in + + da cui si ricava che: + + \[ \dim (W^\perp)^\perp = \dim W + \dim V^\perp - \dim (W \cap V^\perp) = \dim (W + V^\perp). \] + + Dal momento che vale un'inclusione e l'uguaglianza dimensionale, si conclude che $(W^\perp)^\perp = W + V^\perp$, + da cui la tesi. + \end{proof} + + \begin{exercise} Sia $A \in M(n, \CC)$ anti-hermitiana (i.e.~$A = -A^*$). Si dimostri allora che $A$ + è normale e che ammette solo autovalori immaginari. + \end{exercise} + + \begin{proof}[Soluzione] + Si mostra facilmente che $A$ è normale. Infatti $A A^* = A(-A) = -A^2 = (-A)A = A^* A$. Sia allora + $\lambda \in \CC$ un autovalore di $A$ e sia $\v \neq \vec 0$, $\v \in V_\lambda$. Si consideri il prodotto hermitiano + standard $\varphi$ su $\CC^n$. Allora vale la seguente identità: + \begin{gather*} + \lambda \, \varphi(\v, \v) = \varphi(\v, \lambda \v) = \varphi(\v, A \v) = \varphi(A^* \v, \v) = \\ + \varphi(-A\v, \v) = \varphi(-\lambda \v, \v) = -\conj{\lambda} \, \varphi(\v, \v). + \end{gather*} + + Dacché $\varphi$ è definito positivo, $\varphi(\v, \v) \neq 0 \implies \lambda = -\conj{\lambda}$. Allora + $\Re(\lambda) = \frac{\lambda + \conj{\lambda}}{2} = 0$, e quindi $\lambda$ è immaginario, da cui la tesi. + \end{proof} \end{document} \ No newline at end of file diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 5c22b36..b74fac3 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -160,6 +160,7 @@ \newcommand{\w}{\vec{w}} \newcommand{\U}{\vec{u}} \newcommand{\ww}[1]{\vec{w_{#1}}} +\newcommand{\uu}[1]{\vec{u_{#1}}} \newcommand{\mapstoby}[1]{\xmapsto{#1}} \newcommand{\oplusperp}{\oplus^\perp}