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Si consideri ogni orlato $M_j$ di $M$ ottenuto scegliendo + la $j$-esima colonna di $A$: per ipotesi $\det(M_j) = 0$, ed il rango è almeno + $k$. Quindi $\rg(M_j) = k$; poiché le prime $k$ righe sono linearmente indipendenti, l'ultima riga aggiunta deve certamente appartenere al loro + sottospazio generato. Quindi ogni riga di $A^{1,\ldots, k, j}$ appartiene + al sottospazio $\Span(A_1, \ldots, A_k)$, da cui si deduce che $\rg(A^{1,\ldots, k, j}) = k$, e quindi che $\rg(A^{1,\ldots,k,j}) = k \implies A^j \in \Span(A^1, \ldots, A^k) \implies \rg(A) = k$. + + + \subsection{Autovalori e diagonalizzabilità} Sia $f \in \End(V)$. Si dice che $\lambda \in \KK$ è un autovalore @@ -1047,7 +1122,6 @@ tale che $f(\vec{v}) = \lambda \vec{v}$, e in tal caso si dice che $\vec{v}$ è un autovettore relativo a $\lambda$. Un autovalore è tale se esiste una soluzione non nulla a $(f - \lambda \Idv) \vec{v} = \vec{0}$, ossia se e solo se: - \[\det(f - \lambda \Idv) = 0. \] Questa relazione è ben definita dacché il determinante è invariante @@ -1055,7 +1129,7 @@ di $f$. Si definisce allora $p_f(\lambda) = \det(f - \lambda \Idv)$, detto polinomio caratteristico di $f$, ancora invariante per matrici associate a $f$. Si denota inoltre con - spettro di $f$ l'insieme $sp(f)$ degli autovalori di $f$ e con + spettro di $f$ l'insieme $\Sp(f)$ degli autovalori di $f$ e con $V_\lambda = \Ker(f - \lambda \Idv)$ lo spazio degli autovettori relativo a $\lambda$, detto autospazio di $\lambda$. @@ -1075,17 +1149,17 @@ \item il termine noto di $p_f(\lambda)$ è $\det(f - 0 \cdot \Idv) = \det(f)$, \item poiché $p_f(\lambda)$ appartiene all'anello euclideo $\KK[\lambda]$, che è dunque un UFD, esso ammette al più $n$ radici, - \item $sp(f)$ ha al più $n$ elementi, ossia esistono al massimo + \item $\Sp(f)$ ha al più $n$ elementi, ossia esistono al massimo $n$ autovalori (dalla precedente considerazione), \item se $\KK = \CC$ e $\charpoly{f} \in \RR[\lambda]$, $\lambda \in - sp(f) \iff \overline{\lambda} \in sp(f)$ (infatti $\lambda$ è + \Sp(f) \iff \overline{\lambda} \in \Sp(f)$ (infatti $\lambda$ è soluzione di $\charpoly{f}$, e quindi anche $\overline{\lambda}$ deve esserne radice, dacché i coefficienti di $\charpoly{f}$ sono in $\RR$), \item se $\KK$ è un campo algebricamente chiuso, $p_f(\lambda)$ ammette sempre almeno un autovalore distinto (o esattamente $n$ se contati con molteplicità), - \item $0 \in sp(f) \iff \dim \Ker f > 0 \iff \rg f < 0 \iff \det(f) = 0$, + \item $0 \in \Sp(f) \iff \dim \Ker f > 0 \iff \rg f < 0 \iff \det(f) = 0$, \item autovettori relativi ad autovalori distinti sono sempre linearmente indipendenti, \item dati $\lambda_1$, ..., $\lambda_k$ autovalori di $f$, @@ -1103,8 +1177,8 @@ minore di $\mu_a(\lambda)$), \item vale sempre la disuguaglianza $n \geq \sum_{i=1}^k \mu_a(\lambda_i) \geq \sum_{i=1}^k \mu_g(\lambda_i)$, \item se $W \subseteq V$ è un sottospazio $f$-invariante, - allora $\charpolyrestr{f}{W} \mid p_f(\lambda)$\footnote{lavorando - su endomorfismi, la notazione $\restr{f}{W}$ è impiegata per + allora $\charpolyrestr{f}{W} \mid p_f(\lambda)$\footnote{Quando si lavora + su degli endomorfismi, la notazione $\restr{f}{W}$ è impiegata per considerare $f$ ristretta a $W$ sia sul dominio che sul codominio.} (è sufficiente prendere una base di $W$ ed estenderla a base di $V$, considerando poi la matrice associata in tale base, che è a blocchi), @@ -1142,7 +1216,10 @@ che $f$ è diagonalizzabile se e solo se per ogni autovalore la massima taglia di un blocco di Jordan è esattamente $1$, ossia se il polinomio minimo di $f$ è un prodotto di fattori lineari - distinti. + distinti. Si può fare la stessa considerazione guardando al + teorema di decomposizione primaria (gli indici di Fitting del + sottospazio generalizzato sono esattamente le moltiplicità algebriche + degli autovalori nel polinomio minimo). Data $f$ diagonalizzabile, la matrice diagonale $J$ a cui $f$ è associata è, dati gli autovalori $\lambda_1$, ..., $\lambda_k$, @@ -1170,11 +1247,10 @@ Se $f$ è diagonalizzabile, allora ogni spazio $W$ $f$-invariante di $V$ è tale che: - \[ W = (W \cap V_{\lambda_1}) \oplus \cdots \oplus (W \cap V_{\lambda_k}), \] dove $\lambda_1$, ..., $\lambda_k$ sono gli autovalori distinti di - $f$. + $f$, e dunque $\restr{f}{W}$ è sempre diagonalizzabile, se $f$ lo è. Due endomorfismi $f$, $g \in \End(V)$ diagonalizzabili si dicono simultaneamente diagonalizzabili se esiste una base $\basis$ di $V$ tale per cui sia la matrice associata di $f$ in $\basis$ che quella @@ -1191,6 +1267,12 @@ anche base di autovettori di $V_{\lambda_i}$; unendo tutti questi autovettori in un'unica base $\basis$ di $V$, si otterrà dunque che una base in cui le matrici associate di $f$ e $g$ sono diagonali. + + Se $f$ è diagonalizzabile, anche $f^k$ lo è, per ogni $k \in \NN$. Se + ogni vettore di $V$ è un autovettore di $f$, allora $f = \lambda \Id$, + con $\lambda \in \KK$ (è sufficiente considerare l'eventuale esistenza di più + autospazi e due vettori $\v$ e $\w$ di due autospazi distinti e considerare + le due scritture possibili di $f(\v + \w)$). \subsection{Prodotto scalare e congruenza} Si consideri una mappa $\varphi : V \times V \to \KK$. Si dice che @@ -1338,7 +1420,6 @@ Data una base $\basis$ di $V$, se $\abs{\CI(\varphi) \cap \basis} \leq 1$ (ossia se ogni vettore di $\basis$ è anisotropo o al più vi è un vettore isotropo, posto in fondo come $\vv n$), si può trovare una base ortogonale $\basis' = \{ \vv 1', \ldots, \vv n' \}$ a partire da $\basis$ tale che ne mantenga la stessa bandiera, ossia tale che: - \[ \Span(\vv 1', \ldots, \vv i') = \Span(\vv 1, \ldots, \vv i) \forall 1 \leq i \leq n. \] Si definisce $C(\w, \v) = \frac{\varphi(\v, \w)}{\varphi(\w, \w)}$ come il coefficiente di Fourier