From 5d0d61f36f06a285acc4d3867c38fd22fe50ce53 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 30 Jun 2023 20:05:16 +0200 Subject: [PATCH] feat(geometria/schede): aggiunge definizioni di operatori simmetrici, ortogonali, hermitiani e unitari --- Geometria 1/Scheda riassuntiva/main.pdf | 1395 ++++++++++++----------- Geometria 1/Scheda riassuntiva/main.tex | 32 +- 2 files changed, 743 insertions(+), 684 deletions(-) diff --git a/Geometria 1/Scheda riassuntiva/main.pdf b/Geometria 1/Scheda riassuntiva/main.pdf index 32bb601..ef95694 100644 --- a/Geometria 1/Scheda riassuntiva/main.pdf +++ b/Geometria 1/Scheda riassuntiva/main.pdf @@ -439,43 +439,37 @@ endobj endobj 196 0 obj << -/Length 7612 +/Length 7933 /Filter /FlateDecode >> stream -x]Y䶑~_pMܤڈH#GGlXӢ"[uhm,խyf@"痉&El~ů_3yFEHLݼ|Ӿ~6Y}Sz+Zek}^,o|2oˎbSoatvTD߬ !owyU64]}o9UYw=-YdOqM(Ͱ/"tPd&? sW$=],bdєgϚ'8 G{wkdD{ ?!ɡ2[R?oNz$꽙qby#_YSB#uq)kxSӭHWYWz#>>4Oi - ->P. 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Si elencano adesso i tipi principali di matrici: - \begin{itemize} \item $A$ è simmetrica $\defiff A^\top = A$, \item $A$ è antisimmetrica $\defiff A^\top = -A$, @@ -480,6 +479,13 @@ \item $A$ è unitaria ($A \in U(n)$ o $U_n$) $\defiff AA^* = A^*A = I_n$. \end{itemize} + Se $A \in M(m, n, \RR)$, allora $\Ker A^\top A = \Ker A$. Infatti, se $\vec x \in \Ker A^\top A$, allora $A^\top A \vec x = \vec 0 \implies \vec x ^\top A^\top A \vec x = \vec 0 \implies q(A \vec x) = \vec 0 \implies A \vec x = \vec 0 \implies \vec x \in \Ker A$, dove $q$ è la forma quadratica derivante dal prodotto scalare standard + di $\RR^n$. Da questo risultato si deduce anche che $\rg(A^\top A) = \rg(A)$. \\ + \vskip 0.05in + + Se $A \in M(m, n, \CC)$, allora $\Ker A^* A = \Ker A$ e $\rg (A^* A) = \rg(A)$ (si segue la stessa linea + di dimostrazione di sopra). + \subsection{Rango di una matrice} Si definisce rango di una matrice $A$ il numero di colonne linearmente @@ -1843,7 +1849,7 @@ dove $r = \rg(\varphi)$ (teorema di Sylvester, caso complesso; si consideri una base ortogonale e se ne normalizzino i vettori anisotropi), - \item Due matrici simmetriche con stesso rango allora non solo sono SD-equivalenti, ma sono + \item Due matrici simmetriche ad elementi complessi con stesso rango allora non solo sono SD-equivalenti, ma sono anche congruenti, \item (se $\KK = \RR$) Esiste sempre una base ortogonale $\basis$ di $V$ tale che: @@ -2075,16 +2081,32 @@ Se $\varphi$ è un prodotto scalare, $f^* = f^\top$ si chiama \textit{trasposto} di $f$, mentre se $\varphi$ è hermitiano $f^*$ si dice \textit{aggiunto} di $f$. + + D'ora in poi si intenderà con $f^\top$ il trasposto di $f$ (con $\varphi$ scalare) e con $f^*$ l'aggiunto di $f$ (con $\varphi$ hermitiano). \\ + + Un'operatore $f$ si dice \textit{simmetrico} se $f = f^\top$ e quindi se $\varphi(\v, f(\w)) = \varphi(f(\v), \w)$ (analogamente un'operatore si dice \textit{hermitiano} se $f = f^*$). \\ + + Un'operatore $f$ si dice \textit{ortogonale} se è un'isometria da + $(V, \varphi)$ in $(V, \varphi)$, ossia se e solo se $\varphi(\v, \w) = \varphi(f(\v), f(\w))$ (analogamente un'operatore si dice \textit{unitario} se è un'isometria con + $\varphi$ prodotto hermitiano). \\ - D'ora in poi si intenderà con $f^\top$ il trasposto di $f$ (con $\varphi$ scalare) e con $f^*$ l'aggiunto di $f$ (con $\varphi$ hermitiano). + Sia $\basis$ una base di $V$. \begin{itemize} + \item se $\basis$ è una base ortonormale, $M_\basis(f^\top) = M_\basis(f)^\top$ (infatti in tal caso $M_\basis(\varphi) = I_n$), + \item se $\basis$ è una base ortonormale, $M_\basis(f^*) = M_\basis(f)^*$ (come sopra), + \item $f$ è simmetrico $\iff$ $f = f^\top$ $\iff$ $M_\basis(\varphi)\inv M_\basis(f)^\top M_\basis(\varphi) = M_\basis(f^\top) = M_\basis(f)$ $\iff$ $M_\basis(\varphi) M_\basis(f)$ è simmetrica, + \item se $\basis$ è una base ortonormale, $f$ è simmetrico $\iff$ $f = f^\top$ $\iff$ $M_\basis(f) = M_\basis(f)^\top$ $\iff$ $M_\basis(f)$ è simmetrica, + \item $f$ è hermitiano $\iff$ $f = f^*$ $\iff$ $M_\basis(\varphi)\inv M_\basis(f)^* M_\basis(\varphi) = M_\basis(f^*) = M_\basis(f)$ $\iff$ $M_\basis(\varphi) M_\basis(f)$ è hermitiana, + \item se $\basis$ è una base ortonormale, $f$ è hermitiana $\iff$ $f = f^*$ $\iff$ $M_\basis(f) = M_\basis(f)^*$ $\iff$ $M_\basis(f)$ è hermitiana, \item $(f^\top)^\top = f$, \item $(f^*)^* = f$, \item $(\lambda f)^\top = \lambda f^\top$, \item $(\lambda f)^* = \conj{\lambda} f^*$, \item $(f + g)^\top = f^\top + g^\top$, \item $(f + g)^* = f^* + g^*$, + \item $(f \circ g)^\top = g^\top \circ f^\top$, + \item $(f \circ g)^* = g^* \circ f^*$, \item se $f$ è invertibile, $(f^\top)\inv = (f\inv)^\top$ (è sufficiente mostrare che $\varphi((f^\top \circ (f\inv)^\top)(\v), \w) = \varphi(\v, \w)$ e dedurre, sottraendo i due membri, che deve valere $f^\top \circ (f\inv)^\top = \Idv$), \item se $f$ è invertibile, $(f^*)\inv = (f\inv)^*$ (come sopra), @@ -2094,7 +2116,7 @@ \item $\Im f^* = (\Ker f)^\perp$, \item se $W$ è un sottospazio di $V$, allora $W$ è $f$-invariante se e solo se $W^\perp$ è $f^*$-invariante, - \item l'operatore $\top \in \End(\End(V))$ è sempre diagonalizzabili + \item l'operatore $\top \in \End(\End(V))$ è sempre diagonalizzabile e ha spettro $\{\pm 1\}$, dal momento che il suo polinomio minimo è $x^2-1$ (infatti $(f^\top)^\top = f$ e gli autospazi relativi a $1$ e $-1$ sono entrambi diversi da $\zerovecset$), \item l'autospazio $V_1$ di $\top$ raccoglie gli operatori simmetrici, mentre $V_{-1}$ raccoglie gli operatori antisimmetrici. @@ -2253,7 +2275,7 @@ Chiamiamo $D_0$ \textit{direzione} del sottospazio affine $D$. Inoltre $D_0$ è unico e possiamo scriverlo anche come $D_0=\{Q-P\mid P,Q\in D\}$. - In generale i sottospazi affini corrispondo ai traslati di sottospazi vettoriali di $V$ + In generale i sottospazi affini corrispondono ai traslati di sottospazi vettoriali di $V$ Chiamiamo \textit{dimensione} di un sottospazio affine $D$ la dimensione dello spazio vettoriale $D_0$. In particolare dim$E$=dim$V$. Quindi così come per gli spazi vettoriali i sottospazi affini di dimensione 0 sono i punti di $E$, quelli di dimensione 1 sono le rette di $E$, quelli di dimensione 2 sono i piani di $E$ e quelli di codimensione 1 sono gli iperpiani affini.