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+43,11 @@ $\Ker f^k \subseteq \Ker f^{k-1}$, violando la minimalità di $k$, \Lightning. \\ \li Dal momento che vale la decomposizione di Fitting e che $\varphi_{\restr{f}{\Ker f^k}}$ e $\varphi_{\restr{f}{\Im f^k}}$ sono coprimi tra loro (il primo è diviso solo da $t$, mentre il secondo non è diviso da $t$), $\varphi_f = \mcm(\varphi_{\restr{f}{\Ker f^k}}, \varphi_{\restr{f}{\Im f^k}}) = \varphi_{\restr{f}{\Ker f^k}} \varphi_{\restr{f}{\Im f^k}}$. Si conclude quindi che $k = \mu'_a(0)$ rispetto a $\varphi_f$, ossia la molteplicità algebrica di $0$ in tale polinomio. Analogamente si osserva che $t = \mu_a(0)$ rispetto a $p_f$, ossia la molteplicità algebrica - dell'autovalore $0$ in $f$, e quindi che $\mu_a(0) \geq k$. + dell'autovalore $0$ in $f$, e quindi che $\mu_a(0) \geq k$, \\ + \li Considerando l'endomorfismo $g = f - \lambda \Id$, si osservano facilmente alcune analogie tra le proprietà + determinanti di $g$ e di $f$: $p_g(t) = \det(f - \lambda \Id - t \Id) = \det(f - (\lambda + t) \Id) = p_f(\lambda + t) + \implies \mu_{a, g}(0) = \mu_{a, f}(\lambda)$. Si possono dunque riscrive i precedenti risultati in termini delle + molteplicità di un generico autovalore di $f$ considerando la molteplicità di $0$ in $g$. \end{remark} Reiterando la decomposizione di Fitting (o applicando il teorema di decomposizione primaria), si ottiene @@ -56,8 +60,50 @@ \[ V = \Ker (f - \lambda_1 \Id)^{n_1} \oplus \cdots \oplus \Ker (f - \lambda_m \Id)^{n_m}. \] - Si deduce da questa identità che $f$ è diagonalizzabile se e solo se $n_i = 1$ $\forall i \leq m$. - %TODO: aggiungere come proposizione e approfondire + \begin{definition} + Si definisce \textbf{autospazio generalizzato} relativo all'autovalore $\lambda_i$ di $f$, lo spazio: + + \[ \widetilde{V_{\lambda_i}} = \Ker (f - \lambda_i \Id)^{\mu_{a, f}(\lambda_i)} = \Ker (f - \lambda_m \Id)^{n_m}. \] + \end{definition} + + \begin{remark} Riguardo alla decomposizione primaria di $V$ e agli autospazio generalizzati di $f$ si possono fare alcune osservazioni aggiuntive. \\ + + \li Si può riscrive la decomposizione primaria di $V$ in termini degli autospazi generalizzati di $f$ come $V = \oplus_{i=1}^m \widetilde{V_{\lambda_i}}$. \\ + \li Vale in particolare che $\widetilde{V_{\lambda_i}} = \{ \v \in V \mid \exists k \in \NN \mid (f-\lambda_i \Id)^k(\v) = \vec{0} \} = \bigcup_{k=0}^{\infty} \Ker (f - \lambda_i \Id)^k$, tenendo in conto la decomposizione + di Fitting e la minimalità di $n_i$. \\ + \li Considerando la traslazione vista nell'ultima osservazione, si deduce che $\Ker(f - \lambda_i \Id)^{n_i}$ ammette + come unico autovalore $\lambda_i$ (separazione degli autovalori). \\ + \li Poiché $f$ è diagonalizzabile se e solo se $V = \bigoplus_{i=1}^m \Ker(f - \lambda_i \Id)$, si può dedurre + un altro criterio per la diagonalizzabilità, ossia $f$ diagonalizzabile $\impliedby n_i = 1$ $\forall i \leq m$. \\ + \li Del precedente criterio vale anche il viceversa: se $f$ è diagonalizzabile e $\lambda_1$, ..., $\lambda_k$ sono + i suoi autovalori, $V$ ammette una base di autovettori; dati allora gli indici $i_p$ che associano ogni vettore + $\vv p$ all'indice del suo rispettivo autovalore, allora sia + $\vv 1 ^{(\lambda_{i_1})}$, ..., $\vv n ^{(\lambda_{i_n})}$ una base di $V$. Poiché $q(t) = \prod_{i=1}^k (t - \lambda_i)$ è tale che $q(f)$ si annulla in ogni vettore della base e ogni suo fattore lineare è composto da un autovalore di $f$ ed è distinto, deve valere che $\varphi_f = q$. + \end{remark} + + \begin{exercise} + Si calcoli il polinomio minimo di $A = \Matrix{0 & -2 & 0 & -2 & 1 \\ 1 & 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 1 & 2 & 0 & 2 & 0}$. + \end{exercise} + + \begin{solution} + Innanzitutto, si calcola il polinomio caratteristico di $A$, ossia $p_A(t) = (1-t)^3 (1+t)^2$, da cui si ricava + che gli autovalori di $A$ sono $1$ e $-1$, con $\mu_a(1) = 3$ e $\mu_a(-1) = 2$. Si può dunque decomporre $V$ + come: + + \[ V = \Ker (A - I)^3 \oplus \Ker (A + I)^2, \] + + \vskip 0.05in + + e $\varphi_A$ sarà della forma $\varphi_A(t) = (t-1)^{n_1} (t+1)^{n_2}$ con $n_1 \leq 3$ e $n_2 \leq 2$. + \begin{enumerate}[(i)] + \item $\rg(A - I) = 3 \implies \dim \Ker (A - I) = 2 < 3 = \mu_a(-1)$. Si controlli adesso il rango di + $(A-I)^2$: $\rg(A - I)^2 = 2 \implies \dim \Ker (A - I)^2 = 3 = \mu_a(1)$, da cui $n_1 = 2$. + \item $\rg(A + I) = 3 \implies \dim \Ker (A + I) = 2$. Allora, poiché $\dim \Ker (A + I) = 2 = \mu_a(-1)$, + si conclude che $n_2 = 1$. + \end{enumerate} + + Quindi $\varphi_A(t) = (t-1)^2(t+1)$. + \end{solution} \begin{exercise} Sia $A \in M(n, \CC)$ invertibile. Dimostrare allora che se $A^3$ è diagonalizzabile, anche $A$ lo è. @@ -68,7 +114,37 @@ dove $m$ è il numero di autovalori distinti di $A^3$. Allora, detto $p(t) = \prod_{i=1}^m (t^3 - \lambda_i)$, vale che $p(A) = 0$, ossia che $\varphi_A \mid p$. Dal momento che $A$ è invertibile, anche $A^3$ lo è, e quindi $\lambda_i \neq 0$ $\forall i \leq m$. Poiché $p$ è allora fattorizzato in soli termini lineari distinti, - anche $\varphi_A$ deve esserlo, e quindi $A$ deve essere diagonalizzabile. + anche $\varphi_A$ deve esserlo, e quindi $A$ deve essere diagonalizzabile. \\ \end{solution} + \vskip 0.1in + + Nello studio della forma canonica di Jordan è rilevante costruire una base a bandiera tale per cui la matrice + associata in tale base sia una matrice a blocchi diagonale formata da blocchi di Jordan. Si consideri + allora $g = f - \lambda \Id$, e sia $k$ la molteplicità algebrica di $\lambda$ nel polinomio minimo + di $f$ (i.e.~il $k$ minimo già visto precedentemente nella decomposizione di Fitting di $g$). \\ + + Si possono allora definire dei sottospazi $U_i$ secondo le seguenti decomposizioni: + + \begin{align*} + \Ker g^k &= \Ker g^{k-1} \oplus U_1, \\ + \Ker g^{k-1} &= \Ker g^{k-2} \oplus g(U_1) \oplus U_2, \\ + \vdots & \qquad \vdots \qquad \vdots \qquad \vdots \qquad \vdots \\ + \Ker g &= \underbrace{\Ker g^0}_{= \{ \vec 0\}} \oplus \, g^{k-1}(U_1) \oplus \cdots \oplus U_{k}. + \end{align*} + + Si noti che $g$ mantiene la dimensione di $U_i$ ad ogni passo fino a $k-i$ composizioni di $g$ (infatti + $\Ker g^{k-i} \cap U_i \subseteq \Ker g^{k-1} \cap U_i = \{\vec 0\}$, per costruzione dei sottospazi + supplementari $U_i$). In particolare, $\dim U_i = m_i$ rappresenta il numero di blocchi di Jordan + relativi a $\lambda$ di taglia $k-i+1$, e quindi valgono le seguenti identità: + + \begin{align*} + \dim \Ker g^k &= \dim \Ker g^{k-1} + m_1 = \mu_a(\lambda), \\ + \dim \Ker g^{k-1} &= \dim \Ker g^{k-2} + m_1 + m_2, \\ + \vdots & \qquad \vdots \qquad \vdots \qquad \vdots \qquad \vdots \\ + \dim \Ker g &= m_1 + m_2 + \ldots + m_k = \mu_g(\lambda). + \end{align*} + + %TODO: aggiungere osservazioni su come trovare una base di Ker g^j seguendo i blocchi di Jordan. + \end{document} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 4425a0e..ba6d93e 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -84,6 +84,9 @@ #1\arrowvert_{#2} } +\newcommand{\eigsp}[1]{V_{\lambda_{#1}}} +\newcommand{\gensp}[1]{\widetilde{V_{\lambda_{#1}}}} + \DeclareMathOperator{\val}{val} \DeclareMathOperator{\Span}{Span} \newcommand{\charpoly}[1]{p_{#1}}