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1/Teoria spettrale degli endomorfismi/2023-03-22, Forma canonica di Jordan reale e prodotto scalare/main.tex +++ b/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-22, Decomposizione di Jordan, forma canonica di Jordan reale e prodotto scalare/main.tex @@ -11,10 +11,128 @@ \maketitle \begin{center} - \Large \textbf{Forma canonica di Jordan reale e prodotto scalare} + \Large \textbf{Decomposizione di Jordan, forma canonica di Jordan reale e prodotto scalare} \end{center} - %TODO: aggiungere la parte relativa alla forma canonica di Jordan reale + \begin{note} + Nel corso del documento, qualora non specificato, per $f$ si intenderà un qualsiasi + endomorfismo di $V$, dove $V$ è uno spazio vettoriale di dimensione $n \in \NN$. Inoltre + per $\KK$ si intenderà, per semplicità, un campo algebricamente chiuso; altrimenti + è sufficiente considerare un campo $\KK$ in cui i vari polinomi caratteristici esaminati + si scompongono in fattori lineari. + \end{note} + + Sia $J$ la forma canonica di Jordan relativa a $f \in \End(V)$ in una base $\basis$. Allora è possibile decomporre + tale matrice in una somma di due matrici $D$ e $N$ tali che: + + \begin{itemize} + \item $D$ è diagonale e in particolare contiene tutti gli autovalori di $J$; + \item $N$ è nilpotente ed è pari alla matrice ottenuta ignorando la diagonale di $J$; + \item $DN = ND$, dacché le due matrici sono a blocchi diagonali. + \end{itemize} + + Pertanto è possibile considerare gli endomorfismi $\delta = M_\basis\inv(D)$ (diagonalizzabile) e $\nu = M_\basis\inv(N)$ (nilpotente). + Si osserva allora che questi endomorfismi sono tali che $f = \delta + \nu$ (\textbf{decomposizione di Jordan} di $f$). + + \begin{theorem} + La decomposizione di Jordan di $f$ è unica. + \end{theorem} + + \begin{proof} + Per dimostrare che la decomposizione di Jordan è unica è sufficiente mostrare che, dati $\delta$, $\delta'$ + diagonalizzabili e $\nu$, $\nu'$ nilpotenti tali che $f = \delta + \nu = \delta' + \nu'$, deve valere + necessariamente che $\delta = \delta'$ e che $\nu = \nu'$. In particolare è sufficiente dimostrare + che $\restr{\delta}{\Gensp} = \restr{\delta'}{\Gensp}$ per ogni autovalore $\lambda$ di $f$, dal momento + che $V = \gensp 1 \oplus \cdots \oplus \gensp k$, dove $k$ è il numero di autovalori distinti di $f$, e + così le matrici associate dei due endomorfismi sarebbero uguali in una stessa base, da cui si concluderebbe che + $\delta = \delta'$, e quindi che $\nu = \nu'$. \\ + + Si osserva innanzitutto che $\delta$ (e così tutti gli altri tre endomorfismi) commuta con $f$: + $\delta \circ f = \delta \circ (\delta + \nu) \underbrace{=}_{\delta \circ \nu = \nu \circ \delta} (\delta + \nu) \circ \delta = f \circ \delta$. + Da quest'ultimo risultato consegue che $\Gensp$ è $\delta$-invariante, dacché se $f$ commuta con $\delta$, + anche $(f - \lambda \Id)^n$ commuta con $\delta$. Sia infatti + $\v \in \Gensp = \Ker (f - \lambda \Id)^n$, allora $(f - \lambda \Id)^n(\delta(\v)) = \delta((f - \lambda \Id)^n(\v)) = \delta(\vec 0) = \vec 0 \implies \delta(\Gensp) \subseteq \Gensp$. \\ + + Si considerano allora gli endomorfismi $\restr{\delta}{\Gensp}$, $\restr{\delta'}{\Gensp}$, $\restr{\nu}{\Gensp}$, $\restr{\nu'}{\Gensp} \in \End(\Gensp)$. Dal momento che $\restr{\delta}{\Gensp}$ + e $\restr{\nu}{\Gensp}$ commutano, esiste una base $\basis'$ di $\Gensp$ tale per cui i due endomorfismi + sono triangolarizzabili simultaneamente. Inoltre, dal momento che $\restr{\delta}{\Gensp}$ è una restrizione + su $\delta$, che è diagonalizzabile per ipotesi, anche quest'ultimo endomorfismo è diagonalizzabile; + analogamente $\restr{\nu}{\Gensp}$ è ancora nilpotente. \\ + + Si osserva dunque che $M_{\basis'}(\restr{f}{\Gensp}) = + M_{\basis'}(\restr{\delta}{\Gensp}) + M_{\basis'}(\restr{\nu}{\Gensp})$: la + diagonale di $M_\basis'(\restr{\nu}{\Gensp})$ è nulla, e $M_{\basis'}(\restr{f}{\Gensp})$, poiché somma + di due matrici triangolari superiori, è una matrice triangolare superiore. Allora la diagonale di + $M_{\basis'}(\restr{f}{\Gensp})$ raccoglie l'unico autovalore $\lambda$ di $\restr{f}{\Gensp}$, che dunque è + l'unico autovalore anche di $\restr{\delta}{\Gensp}$. In particolare, poiché $\restr{\delta}{\Gensp}$ è + diagonalizzabile, vale che $\restr{\delta}{\Gensp} = \lambda \Id$. Analogamente $\restr{\delta'}{\Gensp} = \lambda \Id$, e quindi $\restr{\delta}{\Gensp} = \restr{\delta'}{\Gensp}$, da cui anche + $\restr{\nu}{\Gensp} = \restr{\nu'}{\Gensp}$. Si conclude dunque che le coppie di endomorfismi sono + uguali su ogni restrizione, e quindi che $\delta = \delta'$ e $\nu = \nu'$. + \end{proof} + + Sia adesso $V = \RR^n$. Si consideri allora la forma canonica di Jordan di $f$ su $\CC$ (ossia estendendo, qualora + necessario, il campo a $\CC$) e sia $\basis$ una base di Jordan per $f$. + Sia $\alpha$ un autovalore di $f$ in $\CC \setminus \RR$. Allora, dacché $p_f \in \RR[\lambda]$, anche + $\conj \alpha$ è un autovalore di $f$. In particolare, vi è un isomorfismo tra $\genspC \alpha$ e $\genspC{\conj{\alpha}}$ (rappresentato proprio dall'operazione di coniugio). Quindi i blocchi di Jordan + relativi ad $\alpha$ e ad $\conj \alpha$ sono gli stessi, benché coniugati. \\ + + Sia ora $\basis'$ una base ordinata di Jordan per $\restr{f}{\genspC \alpha}$, allora $\conj{\basis'}$ è anch'essa una base ordinata di Jordan per $\restr{f}{ \genspC{\conj{\alpha}}}$. Si + consideri dunque $W = \genspC \alpha \oplus \genspC{\conj{\alpha}}$ e la restrizione + $\varphi = \restr{f}{W}$. Si osserva che la forma canonica di $\varphi$ si ottiene estraendo i singoli blocchi relativi + ad $\alpha$ e $\conj \alpha$ dalla forma canonica di $f$. Se $\basis' = \{ \vv 1, ..., \vv k \}$, + si considera $\basis'' = \{ \Re(\vv 1), \imm(\vv 1), ..., \Re(\vv k), \imm(\vv k) \}$, ossia + i vettori tali che $\vv i = \Re(\vv i) + i \imm(\vv i)$. Questi vettori soddisfano due particolari + proprietà: + + \begin{itemize} + \item $\Re(\vv i) = \displaystyle \frac{\vv i + \conj{\vv i}}{2}$, + \item $\imm(\vv i) = \displaystyle \frac{\vv i - \conj{\vv i}}{2i} \underbrace{=}_{\frac{1}{i}=-i} -\frac{\vv i - \conj{\vv i}}{2} i$. + \end{itemize} + + In particolare $\basis''$ è un base di $W$, dal momento che gli elementi di $\basis''$ generano $W$ e sono + tanti quanto la dimensione di $W$, ossia $2k$. Si ponga $\alpha = a + bi$. Se $\vv i$ è autovettore si conclude che:\footnote{Si è in seguito utilizzato più volte l'identità $f(\conj{\vv i}) = \conj{f(\vv i)}$.} + + \begin{itemize} + \item $f(\Re(\vv i)) = \frac{1}{2}\left( f(\vv i) + f( \conj{\vv i}) \right) = + \frac{1}{2}\left( \alpha \vv i + \conj \alpha \conj{\vv i} \right) = + \frac{1}{2}\left( a \vv i + b i \vv i + a \conj{\vv i} - b i \conj{\vv i} \right) + = a \frac{\vv i + \conj{\vv i}}{2} + b \frac{\vv i - \conj{\vv i}}{2} i = + a \Re(\vv i) - b \imm(\vv i)$, + \item $f(\imm(\vv i)) = \frac{1}{2i}\left( f(\vv i) - f( \conj{\vv i}) \right) = + \frac{1}{2i}\left( \alpha \vv i - \conj \alpha \conj{\vv i} \right) = + \frac{1}{2i}\left( a \vv i + b i \vv i - a \conj{\vv i} + b i \conj{\vv i} \right) + = b \frac{\vv i + \conj{\vv i}}{2} + a \frac{\vv i - \conj{\vv i}}{2i} = + b \Re(\vv i) + a \imm(\vv i)$. + \end{itemize} + + Altrimenti, se non lo è: + + \begin{itemize} + \item $f(\Re(\vv i)) = \frac{1}{2}\left( f(\vv i) + f( \conj{\vv i}) \right) = + \frac{1}{2}\left( \alpha \vv i + \vv{i-1} + \conj \alpha \conj{\vv i} + \conj{\vv{i-1}} \right) = + \frac{1}{2}\left( a \vv i + b i \vv i + a \conj{\vv i} - b i \conj{\vv i} \right) + \Re(\vv{i-1}) + = a \frac{\vv i + \conj{\vv i}}{2} + b \frac{\vv i - \conj{\vv i}}{2} i + \Re(\vv{i-1}) = + a \Re(\vv i) - b \imm(\vv i) + \Re(\vv{i-1})$, + \item $f(\imm(\vv i)) = \frac{1}{2i}\left( f(\vv i) - f( \conj{\vv i}) \right) = + \frac{1}{2i}\left( \alpha \vv i + \vv {i-1} - \conj \alpha \conj{\vv i} - \conj{\vv{i-1}} \right) = + \frac{1}{2i}\left( a \vv i + b i \vv i - a \conj{\vv i} + b i \conj{\vv i} \right) + \imm(\vv{i-1}) + = b \frac{\vv i + \conj{\vv i}}{2} + a \frac{\vv i - \conj{\vv i}}{2i} + \imm(\vv{i-1})= + b \Re(\vv i) + a \imm(\vv i) + \imm(\vv{i-1})$. + \end{itemize} + + Quindi la matrice associata nella base $\basis''$ è la stessa di $f$ relativa ad $\alpha$ dove + si amplifica la matrice sostituendo ad $\alpha$ la matrice\footnote{Si verifica facilmente che lo + spazio delle matrici $\left\{\Matrix{a & -b \\ b & a} \in M(2, \RR) \mid a, b \in \RR\right\}$ è isomorfo a $\CC$ + secondo la mappa $\Matrix{a & -b \\ b & a} \mapsto a + bi$.} $\Matrix{a & -b \\ b & a}$ e ad + $1$ la matrice $\Matrix{1 & 0 \\ 0 & 1}$. + + \begin{example} + Si consideri la matrice $M = \Matrix{1+i & 1 & 0 & 0 \\ 0 & 1+i & 0 & 0 \\ 0 & 0 & 1-i & 1 \\ 0 & 0 & 0 & 1-i}$. + Si osserva che $M$ è composta da due blocchi che sono uno il blocco coniugato dell'altro. Quindi + $M$ è simile alla matrice reale $\Matrix{1 & -1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & 1}$. + \end{example} + + \hr \begin{definition} Un \textbf{prodotto scalare} su $V$ è una forma bilineare simmetrica $\varphi$ con argomenti in $V$. diff --git a/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-22, Forma canonica di Jordan reale e prodotto scalare/main.pdf b/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-22, Forma canonica di Jordan reale e prodotto scalare/main.pdf deleted file mode 100644 index 757db67db36e05bd07efd90e34e16d50c58d0e60..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 199907 zcmb@u1z40#_c%;SDxsv5Qqt_gvP%d^N=SD%(jg!vC4wN*DXB=8gh)4nAV@b7A|M@- z65qYM_{j6V@9&N4|8ZRmbI+VPGjsZ!dlyC}2`LT`Cj^(VZ>W6?mxqp<&cVbQS4aq# zOWxex(&Zr?7{t#@hxozelCrXOF?XiplCm{+F_$nmbucr>6&A*Q%Bca?lrjYl$CYAD>R1G 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+++ b/tex/latex/style/personal_commands.sty @@ -23,6 +23,8 @@ \newcommand{\wip}{\begin{center}\textit{Questo avviso sta ad indicare che questo documento è ancora una bozza e non è da intendersi né completo, né revisionato.}\end{center}} +\newcommand\hr{\vskip 0.05in \par\vspace{-.5\ht\strutbox}\noindent\hrulefill\par} + % Modalità matematica/fisica \let\oldvec\vec \renewcommand{\vec}[1]{\underline{#1}} @@ -92,6 +94,9 @@ \newcommand{\RRbar}{\overline{\RR}} % Spesso utilizzati al corso di Geometria 1. +\newcommand{\conj}[1]{\overline{#1}} + +\let\imm\Im \let\Im\undefined \DeclareMathOperator{\Im}{Im} \DeclareMathOperator{\Rad}{Rad} @@ -113,10 +118,11 @@ \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\Cof}{Cof} -\newcommand{\Eigsp}[1]{V_{\lambda}} -\newcommand{\Gensp}[1]{\widetilde{V_{\lambda}}} +\newcommand{\Eigsp}[0]{V_{\lambda}} +\newcommand{\Gensp}[0]{\widetilde{V_{\lambda}}} \newcommand{\eigsp}[1]{V_{\lambda_{#1}}} \newcommand{\gensp}[1]{\widetilde{V_{\lambda_{#1}}}} +\newcommand{\genspC}[1]{\widetilde{V_{#1}}} \DeclareMathOperator{\val}{val} \DeclareMathOperator{\Span}{Span}