From b10ec4b5247efd25917aa81a8cfa04f4b16b1de8 Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Mon, 27 Mar 2023 10:27:59 +0200 Subject: [PATCH] feat(geometria): aggiunge le schede riassuntive --- Geometria I/Scheda riassuntiva/main.pdf | Bin 0 -> 302492 bytes Geometria I/Scheda riassuntiva/main.tex | 1207 +++++++++++++++++ .../main.pdf | Bin 190876 -> 190876 bytes .../main.tex | 0 tex/latex/style/personal_commands.sty | 17 + 5 files changed, 1224 insertions(+) create mode 100644 Geometria I/Scheda riassuntiva/main.pdf create mode 100644 Geometria I/Scheda riassuntiva/main.tex rename Geometria I/Teoria spettrale degli endomorfismi/{2023-03-24 => 2023-03-24, Es. forma canonica di Jordan e autospazi generalizzati}/main.pdf (99%) rename Geometria I/Teoria spettrale degli endomorfismi/{2023-03-24 => 2023-03-24, Es. forma canonica di Jordan e autospazi generalizzati}/main.tex (100%) diff --git a/Geometria I/Scheda riassuntiva/main.pdf b/Geometria I/Scheda riassuntiva/main.pdf new file 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I/Scheda riassuntiva/main.tex b/Geometria I/Scheda riassuntiva/main.tex new file mode 100644 index 0000000..e946773 --- /dev/null +++ b/Geometria I/Scheda riassuntiva/main.tex @@ -0,0 +1,1207 @@ +\documentclass[10pt,landscape]{article} +\usepackage{amssymb,amsmath,amsthm,amsfonts} +\usepackage{personal_commands} +\usepackage{multicol,multirow} +\usepackage{marvosym} +\usepackage{calc} +\usepackage{ifthen} +\usepackage[landscape]{geometry} +\usepackage[colorlinks=true,citecolor=blue,linkcolor=blue]{hyperref} + + +\ifthenelse{\lengthtest { \paperwidth = 11in}} +{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } +{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} + {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } + {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } +} +\pagestyle{empty} +\makeatletter +\renewcommand{\section}{\@startsection{section}{1}{0mm}% + {-1ex plus -.5ex minus -.2ex}% + {0.5ex plus .2ex}%x + {\normalfont\large\bfseries}} +\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% + {-1explus -.5ex minus -.2ex}% + {0.5ex plus .2ex}% + {\normalfont\normalsize\bfseries}} +\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% + {-1ex plus -.5ex minus -.2ex}% + {1ex plus .2ex}% + {\normalfont\small\bfseries}} +\makeatother +\setcounter{secnumdepth}{0} +\setlength{\parindent}{0pt} +\setlength{\parskip}{0pt plus 0.5ex} +% ----------------------------------------------------------------------- + +\title{Schede riassuntive di Geometria 1} + +\begin{document} + + \parskip=0.7ex + + \raggedright + \footnotesize + + \begin{center} + \Large{\textbf{Schede riassuntive di Geometria 1}} \\ + \end{center} + \begin{multicols}{3} + \setlength{\premulticols}{1pt} + \setlength{\postmulticols}{1pt} + \setlength{\multicolsep}{1pt} + \setlength{\columnsep}{2pt} + + \subsection{Alcuni accenni alla geometria di $\RR^3$} + + Si definisce prodotto scalare la forma + bilineare simmetrica unicamente determinata da $\innprod{\vec{e_i}}{\vec{e_j}} = \delta_{ij}$. Vale la seguente identità: $\innprod{(x, y, z)}{(x', y', z')} = xx' + yy' + zz'$. + + Inoltre $\innprod{\vec{a}}{\vec{b}} = \card{\vec{a}} \card{\vec{b}} \cos(\theta)$, dove $\theta$ è l'angolo compreso tra i due vettori. + Due vettori $\vec{a}$, $\vec{b}$ si dicono ortogonali + se e solo se $\innprod{\vec{a}}{\vec{b}} = 0$. + + Si definisce prodotto vettoriale la forma bilineare alternante + da $\RR^3 \times \RR^3$ + in $\RR^3$ tale che $\vec{e_1} \times \vec{e_2} = \vec{e_3}$, + $\vec{e_2} \times \vec{e_3} = \vec{e_1}$, + $\vec{e_3} \times \vec{e_1} = \vec{e_2}$ e + $\vec{e_i} \times \vec{e_i} = \vec{0}$. Dati due + vettori $(x, y, z)$ e $(x', y', z')$, si può determinarne + il prodotto vettoriale informalmente come: + + \[ \begin{vmatrix} + \vec{e_1} & \vec{e_2} & \vec{e_3} \\ + x & y & z \\ + x' & y' & z' + \end{vmatrix} . \] + + Vale l'identità $\card{\vec{a} \times \vec{b}} = \card{\vec{a}} \card{\vec{b}} \sin(\theta)$, dove $\theta$ è l'angolo con cui, ruotando di + $\theta$ in senso antiorario $\vec{a}$, si ricade su $\vec{b}$. + Due vettori $\vec{a}$, $\vec{b}$ si dicono paralleli se $\exists + k \mid \vec{a} = k \vec{b}$, o equivalentemente se + $\vec{a} \times \vec{b} = \vec{0}$. Altrettanto si può dire + se $\innprod{\vec{a}}{\vec{b}} = \card{\vec{a}} \card{\vec{b}}$ (i.e. + $\cos(\theta) = 1 \implies \theta = 0$). + + Una retta in $\RR^3$ è un sottospazio affine della + forma $\vec{v} + \Span(\vec{r})$. Analogamente + un piano è della forma $\vec{v} + \Span(\vec{x}, \vec{y})$. + + Nella forma cartesiana, un piano è della forma $ax+by+cz=d$, + dove $(a,b,c)$ è detta normale del piano. Una retta è + l'intersezione di due piani, e dunque è un sistema lineare + di due equazioni di un piano. Due piani sono perpendicolari + fra loro se e solo se le loro normali sono ortogonali. Due + piani sono paralleli se e solo se le loro normali sono parallele. + Il vettore $\vec{r}$ che genera lo $\Span$ di una retta che è + intersezione di due piani può essere computato come + prodotto vettoriale delle normali dei due piani. + + Valgono le seguenti identità: + + \begin{itemize} + \item $\vec{a} \times (\vec{b} \times \vec{c}) = + \innprod{\vec{a}}{\vec{c}}\,\vec{b} - \innprod{\vec{a}}{\vec{b}}\,\vec{c}$ (\textit{identità di Lagrange}), + \item $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = + \vec{0}$ (\textit{identità di Jacobi}). + \end{itemize} + + Dati tre punti $\vec{a}$, $\vec{b}$, $\vec{c}$, il volume + del parallelepipedo individuato da questi punti è: + + \[\card{\det\begin{pmatrix}\vec{a} \\ \vec{b} \\ \vec{c}\end{pmatrix}} = + \card{\innprod{\vec{a}}{\vec{b} \times \vec{c}}}.\] + + Tre punti sono complanari se e solo se il volume di tale parallelpipedo è nullo + (infatti questo è equivalente a dire che almeno uno dei tre punti + si scrive come combinazione lineare degli altri due). + + + \subsection{Proprietà generali di uno spazio vettoriale} + + Uno spazio vettoriale $V$ su un campo $\KK$ soddisfa i seguenti + assiomi: + + \begin{itemize} + \item $(V, +)$ è un gruppo abeliano, + \item il prodotto esterno da $\KK \times V$ in $V$ è + associativo rispetto agli scalari (i.e. $a(b\vec{v}) = (ab)\vec{v}$), + \item $1_{\KK} \cdot \vec{v} = \vec{v}$, + \item il prodotto esterno è distributivo da ambo i + lati (i.e. $(a+b)\vec{v} = a\vec{v} + b\vec{v}$ e + $a(\vec{v} + \vec{w}) = a\vec{v} + a\vec{w}$. + \end{itemize} + + Un insieme di vettori $I$ si dice linearmente indipendente se + una qualsiasi combinazione lineare di un suo sottinsieme + finito è nulla se e solo se i coefficienti dei vettori + sono tutti nulli. Si dice linearmente dipendente in caso + contrario. + + Un insieme di vettori $G$ si dice generatore di $V$ se ogni vettore + di $V$ si può scrivere come combinazione lineare di un numero + finito di elementi di $G$, ossia se $V = \Span(G)$. + + Una base è un insieme contemporaneamente linearmente indipendente + e generatore di $V$. Equivalentemente una base è un insieme generatore + minimale rispetto all'inclusione e un insieme linearmente indipendente + massimale, sempre rispetto all'inclusione. Ogni spazio vettoriale, + anche quelli non finitamente generati, + ammettono una base. La dimensione della base è unica ed è il + numero di elementi dell'insieme che è base. + + Dato un insieme linearmente indipendente $I$ in uno spazio di dimensione + finita, tale insieme, data una base $\basis$, può essere esteso + a una base $T$ che contiene $I$ e che è completato da + elementi di $\basis$. + + Analogamente, dato un insieme generatore finito $G$, da esso + si può estrarre sempre una base dello spazio. + + Uno spazio vettoriale fondato su un campo infinito + con un insieme di vettori infinito non + è mai unione finita di sottospazi propri. Un insieme linearmente + indipendente di $V$ con esattamente $\dim V$ elementi è una + base di $V$. Analogamente, un insieme generatore di $V$ con esattamente + $\dim V$ elementi è una base di $V$. + + Sia $\basis = \{\vec{v_1}, \ldots, \vec{v_n}\}$ una base ordinata dello spazio vettoriale $V$. + + \begin{itemize} + \item $\zerovecset$ e $V$ sono detti sottospazi banali, + \item lo $\Span$ di $n$ vettori è il più piccolo sottospazio + di $V$ contenenti tali vettori, + \item $\Span(\basis) = V$, + \item $\Span(\emptyset) = \zerovecset$, + \item dato $X$ generatore di $V$, $X \setminus \{\vec{x_0}\}$ + genera $V \iff \vec{x_0} \in \Span(X \setminus \{\vec{x_0}\})$, + \item $X \subseteq Y$ è un sottospazio di $Y \iff \Span(X) = X$, + \item $\Span(X) \subseteq Y \iff X \subseteq Y$, se $Y$ è uno spazio, + \item $\Span(\Span(A)) = \Span(A)$, + \item se $I$ è un insieme linearmente indipendente di $V$, + allora $\card{I} \leq \dim V$, + \item se $G$ è un insieme generatore di $V$, allora + $\card{G} \geq \dim V$, + \item $[\vec{v}]_\basis$ è la rappresentazione + di $\vec{v}$ nella base ordinata $\basis$, ed è + un vettore di $\KK^n$ che alla coordinata $i$-esima + associa il coefficiente di $\vec{v_i}$ nella combinazione + lineare di $\vec{v}$ nella base $\basis$, + \item la rappresentazione nella base $\basis$ è sempre + unica ed esiste sempre (è quindi un isomorfismo tra $V$ e + $\KK^n$), + \item si definisce base canonica di $\KK^n$ la base + $e = \{\vec{e_1}, \ldots, \vec{e_n}\}$, dove + $\vec{e_i}$ è un vettore con tutte le coordinate nulle, + eccetto per la $i$-esima, che è pari ad $1$ (pertanto + $\dim \KK^n = n$), + \item una base naturale di $M(m, n, \KK)$ è data + da $\basis = \{E_{11}, E_{12}, \ldots, E_{1n}, \ldots, E_{mn}\}$, + dove $E_{ij}$ è una matrice con tutti gli elementi nulli, eccetto + quello nel posto $(i, j)$, che è pari ad $1$ (dunque + $\dim M(m, n, \KK) = mn$), + \item le matrici $A$ di taglia $n$ tali che $A^\top = A$ formano il + sottospazio $\Sym(n, \KK)$ di $M(n, \KK)$, detto sottospazio delle matrici + simmetriche, la cui base naturale è data da + $\basis' = \{E_{ij} + E_{ji} \in \basis \mid i < j\} \cup + \{E_{ij} \in \basis \mid i = j\}$, dove $\basis$ è la + base naturale di $M(m, n, \KK)$ (dunque $\dim \Sym(n, \KK) = \frac{n(n+1)}{2}$), + \item le matrici $A$ di taglia $n$ tali che $A^\top = -A$ formano il + sottospazio $\Lambda(n, \KK)$ di $M(n, \KK)$, detto sottospazio delle matrici + antisimmetriche, la cui base naturale è data da + $\basis' = \{E_{ij} - E_{ji} \in \basis \mid i < j\}$, dove $\basis$ è la + base naturale di $M(m, n, \KK)$ (dunque $\dim \Lambda(n, \KK) = \frac{n(n-1)}{2}$), + \item poiché $\Sym(n, \KK) \cap \Lambda(n, \KK) = \zerovecset$ e + $\dim \Sym(n, \KK) + \dim \Lambda(n, \KK) = \dim M(n, \KK)$, + vale che $M(n, \KK) = \Sym(n, \KK) \oplus \Lambda(n, \KK)$, + \item una base naturale di $\KK[x]$ è data da $\basis = \{x^n \mid + n \in \NN \}$, mentre una di $\KK_t[x]$ è data da $\basis \cap + \KK_t[x] = \{x^n \mid n \in \NN \land n \leq t\}$ (quindi + $\dim \KK[x] = \infty$ e $\dim \KK_t[x] = t+1$), + \item una base naturale di $\KK$ è $1_\KK = \{1_\KK\}$ (quindi + $\dim \KK = 1$), + \item un sottospazio di dimensione $1$ si definisce \textit{retta}, + uno di dimensione $2$ \textit{piano}, uno di dimensione $3$ + \textit{spazio}, e, infine, uno di dimensione $n-1$ un iperpiano, + \item un iperpiano $\Pi$ è sempre rappresentabile da un'equazione cartesiana + nelle coordinate della rappresentazione della base (infatti ogni + iperpiano è il kernel di un funzionale $f \in \dual{V}$, e $M^\basis_{1_\KK}(f) \, [\vec{v}]_\basis = 0$ è l'equazione cartesiana; è sufficiente prendere una base di $\Pi$ e completarla + a base di $V$ con un vettore $\vec{t}$, considerando infine + $\Ker \dual{\vec{t}}$). + + \end{itemize} + + \subsection{Applicazioni lineari, somme dirette, quozienti e + prodotti diretti} + + Un'applicazione da $V$ in $W$ si dice applicazione lineare + se: + + \begin{itemize} + \item $f(\vec{v} + \vec{w}) = f(\vec{v}) + f(\vec{w})$, + \item $f(\alpha\vec{v}) = \alpha f(\vec{v})$. + \end{itemize} + + Si definisce $\mathcal{L}(V, W) \subseteq W^V$ come lo spazio delle + applicazioni lineari da $V$ a $W$. Si definisce + $\End(V)$ come lo spazio degli endomorfismi di $V$, ossia + delle applicazioni lineari da $V$ in $V$, dette anche + operatori. Un'applicazione lineare si dice isomorfismo + se è bigettiva. La composizione di funzioni è associativa. + + Dato un sottospazio $A$ di $V$, si definisce lo spazio + quoziente $V/A$ come l'insieme quoziente $V/{\sim}$ della relazione + di equivalenza $\vec{a} \sim \vec{b} \iff a-b \in A$ dotato + dell'usuale somma e prodotto esterno. Si scrive $[\vec{v}]_A$ + come $\vec{v} + A$ e vale che $A = \vec{0} + A$. In particolare + $\vec{v} + A = A \iff \vec{v} \in A$. + + Siano $f : V \to W$, $h : V \to W$, $g : W \to Z$ tre + applicazioni lineari. + $\basis_V$ e $\basis_W$ sono + due basi rispettivamente di $V$ e $W$. In particolare + sia $\basis_V = \{\vec{v_1}, \ldots, \vec{v_n}\}$. Si + ricorda che $\rg(f) = \dim \Im f$. Siano $e$ ed $e'$ le + basi canoniche rispettivamente di $\KK^n$ e $\KK^m$. + + \begin{itemize} + \item $f(\vec{0}_V) = \vec{0}_W$, + \item $\Ker f = f^{-1}(\vec{0}_W)$ è un sottospazio di $V$, + \item $\Im f = f(V)$ è un sottospazio di $W$, + \item $\Im f = \Span(f(\vec{v_1}), \ldots, f(\vec{v_n}))$, + \item $f$ è iniettiva $\iff \Ker f = \zerovecset$, + \item $V/\Ker f \cong \Im f$ (\textit{primo teorema d'isomorfismo}), + \item $\dim \Ker f + \dim \Im f = \dim V$ (\textit{teorema del rango}, o formula delle dimensioni, + valido se la dimensione di $V$ è finita), + \item $g \circ f$ è un'applicazione lineare da $V$ in $Z$, + \item la composizione di funzioni è associativa e distributiva + da ambo i lati, + \item $g \circ (\alpha f) = \alpha (g \circ f) = (\alpha g) \circ f$, + se $\alpha \in \KK$, + \item $\Ker f \subseteq \Ker (g \circ f)$, + \item $\Im (g \circ f) \subseteq \Im g$, + \item $\dim \Im (g \circ f) = \dim \Im \restr{g}{\Im f} = + \dim \Im f - \dim \Ker \restr{g}{\Im f} = \dim \Im f - + \dim (\Ker g \cap \Im f)$ (è sufficiente applicare la formula delle dimensioni sulla composizione), + \item $\dim \Im (g \circ f) \leq \min\{\dim \Im g, \dim \Im f\}$, + \item $\dim \Ker (g \circ f) \leq \dim \Ker g + \dim \Ker f$ (è + sufficiente applicare la formula delle dimensioni su + $\restr{(g \circ f)}{\Ker (g \circ f)}$), + \item $f$ iniettiva $\implies \dim V \leq \dim W$, + \item $f$ surgettiva $\implies \dim V \geq \dim W$, + \item $f$ isomorfismo $\implies \dim V = \dim W$, + \item $g \circ f$ iniettiva $\implies f$ iniettiva, + \item $g \circ f$ surgettiva $\implies g$ surgettiva, + \item $f$ surgettiva $\implies \rg(g \circ f) = \rg(g)$, + \item $g$ iniettiva $\implies \rg(g \circ f) = \rg(f)$, + \item $M^{\basis_V}_{\basis_W}(f) = \begin{pmatrix} \; [f(\vec{v_1})]_{\basis_W} \, \mid \, \cdots \, \mid \, [f(\vec{v_n})]_{\basis_W} \; \end{pmatrix}$ è la matrice + associata a $f$ sulle basi $\basis_V$, $\basis_W$, + \item $M^V_W(f + h) = M^V_W(f) + M^V_W(h)$, + \item $M^V_Z(g \circ f) = M^W_Z(g) M^V_W(f)$, + \item data $A \in M(m, n, \KK)$, sia $f_A : \KK^n \to \KK^m$ tale + che $f_A(\vec{x}) = A \vec{x}$, allora $M^{e}_{e'}(f_A) = A$, + \item $f$ è completamente determinata dai suoi valori in una + qualsiasi base di $V$ ($M^{\basis_V}_{\basis_W}$ è un isomorfismo + tra $\mathcal{L}(V, W)$ e $M(\dim W, \dim V, \mathbb{K})$), + \item $\dim \mathcal{L}(V, W) = \dim V \cdot \dim W$ (dall'isomorfismo + di sopra), + \item $[\,]^{-1}_{\basis_W} \circ M^{\basis_V}_{\basis_W}(f) \circ + {[\,]_{\basis_V}} = f$, + \item $[f(\vec{v})]_{\basis_W} = M^{\basis_V}_{\basis_W}(f) \cdot + [\vec{v}]_{\basis_V}$, + \item $\Im(f) = [\,]^{-1}_{\basis_W}\left(\Im M^{\basis_V}_{\basis_W}(f)\right)$ + \item $\rg(f) = \rg\left(M^{\basis_V}_{\basis_W}(f)\right)$, + \item $\Ker(f) = [\,]^{-1}_{\basis_V}\left(\Ker M^{\basis_V}_{\basis_W}(f)\right)$, + \item $\dim \Ker(f) = \dim \Ker M^{\basis_V}_{\basis_W}(f)$. + \end{itemize} + + Siano $\basis_V'$, $\basis_W'$ altre due basi rispettivamente + di $V$ e $W$. Allora vale il \textit{teorema del cambiamento + di base}: + + \[ M^{\basis_V'}_{\basis_W'}(f) = M^{\basis_W}_{\basis_W'}(id_W) \, + M^{\basis_V}_{\basis_W}(f) \, M^{\basis_V'}_{\basis_V}(id_V).\] + + Siano $A$ e $B$ due sottospazi di $V$. $\basis_A$ e $\basis_B$ sono + due basi rispettivamente di $A$ e $B$. + + \begin{itemize} + \item $A+B = \{\vec{a}+\vec{b} \in V \mid \vec{a} \in A, \vec{b} \in + B\}$ è un sottospazio, + \item $\dim (A+B) = \dim A + \dim B - \dim (A \cap B)$ + (\textit{formula di Grassmann}), + \item $A$ e $B$ sono in somma diretta $\iff A \cap B = \zerovecset \iff$ ogni elemento di $A+B$ si scrive in modo unico come somma di + $\vec{a} \in A$ e $\vec{b} \in B \iff \dim (A+B) = \dim A + \dim B$ + (in tal caso si scrive $A+B = A\oplus B$), + \item $\dim V/A = \dim V - \dim A$ (è sufficiente applicare il + teorema del rango alla proiezione al quoziente), + \item $\dim V \times W = \dim V + \dim W$ ($\basis_V \times \{\vec{0}_W\} \cup \{\vec{0}_V\} \times \basis_W$ è una base + di $V \times W$). + \end{itemize} + + Si definisce \textit{immersione} da $V$ in $V \times W$ + l'applicazione lineare $i_V$ tale che $i_V(\vec{v}) = (\vec{v}, \vec{0})$. + Si definisce \textit{proiezione} da $V \times W$ in $V$ + l'applicazione lineare $p_V$ tale che $p_V(\vec{v}, \vec{w}) = \vec{v}$. + Analogamente si può fare con gli altri spazi del prodotto cartesiano. + + Si dice che $B$ è un supplementare di $A$ se $V = A \oplus B \iff + \dim A + \dim B = \dim V \land A \cap B = \zerovecset$. Il supplementare + non è per forza unico. Per trovare un supplementare di $A$ è sufficiente + completare $\basis_A$ ad una base $\basis$ di $V$ e considerare + $\Span(\basis \setminus \basis_A)$. + + \subsection{Proprietà generali delle matrici} + + Si dice che una matrice $A \in M(n, \KK)$ è singolare se $\det(A) = 0$, + o equivalentemente se non è invertibile. Compatibilmente, si + dice che una matrice $A \in M(n, \KK)$ è non singolare se $\det(A) \neq + 0$, ossia se $A$ è invertibile. + + Si definisce la matrice trasposta di + $A \in M(m, n, \KK)$, detta $A^\top$, in modo + tale che $A_{ij} = A^\top_{ji}$. + + \begin{itemize} + \item $(AB)^\top = B^\top A^\top$, + \item $(A+B)^\top = A^\top + B^\top$, + \item $(\lambda A)^\top = \lambda A^\top$, + \item $(A^\top)^\top = A$, + \item se $A$ è invertibile, $(A^\top)^{-1} = (A^{-1})^\top$, + \item $ \begin{pmatrix} + A + & \rvline & B \\ + \hline + C & \rvline & + D + \end{pmatrix}\begin{pmatrix} + E + & \rvline & F \\ + \hline + G & \rvline & + H + \end{pmatrix}=\begin{pmatrix} + AE+BG + & \rvline & AF+BH \\ + \hline + CE+DG & \rvline & + CF+DH + \end{pmatrix}$. + \end{itemize} + + Siano $A \in M(m, n, \KK)$ e $B \in M(n, m, \KK)$. + + Si definisce $\GL(n, \KK)$ come il gruppo delle matrici + di taglia $n$ invertibili sulla moltiplicazione matriciale. Si definisce + triangolare superiore una matrice i cui elementi al di sotto + della diagonale sono nulli, mentre si definisce triangolare + inferiore una matrice i cui elementi nulli sono quelli al di sopra + della diagonale. + + Si definiscono + \[ Z(M(n, \KK)) = \left\{ A \in M(n, \KK) \mid AB=BA \, \forall B \in M(n, \KK) \right\}, \] + ossia l'insieme delle matrici che commutano con tutte le altre matrici, e + \[ Z_{\GL}(M(n, \KK)) = \left\{ A \in M(n, \KK) \mid AB=BA \, \forall B \in \GL(n, \KK) \right\}, \] + ovvero l'insieme delle matrici che commutano con tutte le matrici + di $\GL(n, \KK)$. + + Si definisce $\tr \in M(m, \KK)^*$ come il funzionale che associa + ad ogni matrice la somma degli elementi sulla sua diagonale. + + \begin{itemize} + \item $\tr(A^\top) = \tr(A)$, + \item $\tr(AB) = \tr(BA)$, + \item $Z(M(n, \KK)) = \Span(I_n)$, + \item $Z_{\GL}(M(n, \KK)) = \Span(I_n)$. + \end{itemize} + + Sia $A \in M(n, \KK)$. Sia $C_A \in \End(M(n, \KK))$ definito in modo + tale che $C_A(B) = AB - BA$. Allora $\Ker C_A = M(n, \KK) + \iff A \in \Span(I_n)$. Siano $I$ un insieme di $n^2$ indici + distinti, allora l'insieme + + \[ T = \left\{ A^i \mid i \in I \right\} \] + + è linearmente dipendente (è sufficiente notare che + se così non fosse, se $A \notin \Span(I_n)$, + tale $T$ sarebbe base di $M(n, \KK)$, ma + così $\Ker C_A = M(n, \KK) \implies A \in \Span(I_n)$, + \Lightning{}, e che se $A \in \Span(I_n)$, $T$ + è chiaramente linearmente dipendente). + + In generale esiste sempre un polinomio $p(X) \in \KK[x]$ + di grado $n$ tale per cui $p(A) = 0$, dove un tale polinomio + è per esempio il polinomio caratteristico di $p$, ossia $p(\lambda)= + \det(\lambda I_n - A)$ (\textit{teorema di + Hamilton-Cayley}). + + \subsection{Rango di una matrice} + + Si definisce rango di una matrice $A$ il numero di colonne linearmente + indipendenti di $A$. Siano $A$, $B \in M(m, n, \KK)$. + + \begin{itemize} + \item $\rg(A) = \rg(A^\top)$ (i.e. il rango è lo stesso se calcolato + sulle righe invece che sulle colonne), + \item $\rg(A) \leq \min\{m, n\}$ (come conseguenza dell'affermazione + precedente), + \item $\rg(A+B) \leq \rg(A) + \rg(B) \impliedby \Im (A+B) \subseteq + \Im(A) + \Im(B)$, + \item $\rg(A+B) = \rg(A) + \rg(B) \implies \Im(A+B) = \Im(A) \oplus \Im(B)$ (è sufficiente applicare la formula di Grassmann), + \item $\rg(A)$ è il minimo numero di matrici di rango uno che + sommate restituiscono $A$ (è sufficiente usare la proposizione + precedente per dimostrare che devono essere almeno $\rg(A)$), + \item $\rg(A)=1 \implies \exists B \in M(m, 1, \KK)$, $C \in M(1, n, \KK) \mid A=BC$ (infatti $A$ può scriversi come $\begin{pmatrix}[c|c|c]\alpha_1 A^i & \cdots & \alpha_n A^i \end{pmatrix}$ per un certo $i \leq n$ tale che $A^i \neq \vec{0}$). + \end{itemize} + + Siano $A \in M(m, n, \KK)$, $B \in M(n, k, \KK)$ e $C \in M(k, t, \KK)$. + + \begin{itemize} + \item $\rg(AB) \geq \rg(A) + \rg(B) - n$ (\textit{disuguaglianza + di Sylvester} -- è sufficiente + usare la formula delle dimensioni ristretta alla composizione + $f_A \circ f_B$), + \item $\rg(ABC) \geq \rg(AB) + \rg(BC) - \rg(B)$ (\textit{disuguaglianza di Frobenius}, di cui la proposizione + precedente è un caso particolare con $B = I_n$ e $k=n$), + \item $\rg(AB) = \rg(B) \impliedby \Ker A = \zerovecset$ (è + sufficiente usare la formula delle dimensioni ristretta + alla composizione $f_A \circ f_B$), + \item $\rg(AB) = \rg(A) \impliedby f_B$ surgettiva (come sopra). + \end{itemize} + + Sia $A \in M(n, \KK)$. + + \begin{itemize} + \item se $A$ è antisimmetrica e il campo su cui si fonda + lo spazio vettoriale non ha caratteristica $2$, allora + $\rg(A)$ è pari, + \item $\rg(A) = n \iff \dim \Ker A = 0 \iff \det(A) \neq 0 \iff A$ è invertibile, + \end{itemize} + + \subsection{Sistemi lineari, algoritmo di eliminazione di Gauss ed + SD-equivalenza} + + Un sistema lineare di $m$ equazioni in $n$ variabili può essere + rappresentato nella forma $A\vec{x} = B$, dove $A \in M(m, n, \KK)$, + $\vec{x} \in \KK^n$ e $B \in \KK^m$. Un sistema lineare si + dice omogeneo se $B = \vec{0}$. In tal caso l'insieme delle soluzioni del + sistema coincide con $\Ker A = \Ker f_A$, dove $f_A : \KK^n \to \KK^m$ è + l'applicazione lineare indotta dalla matrice $A$. Le soluzioni + di un sistema lineare sono raccolte nel sottospazio affine + $\vec{s} + \Ker A$, dove $\vec{s}$ è una qualsiasi soluzione + del sistema completo. + + \begin{itemize} + \item $A\vec{x} = B$ ammette soluzione se e solo se + $B \in \Span(A^1, \ldots, A^n) \iff \Span(A^1, \ldots, A^n, B) = + \Span(A^1, \ldots, A^n) \iff \dim \Span(A^1, \ldots, A^n, B) = + \dim \Span(A^1, \ldots, A^n) \iff + \dim \Im (A \mid B) = \dim \Im A \iff \rg (A \mid B) = \rg (A)$ + (\textit{teorema di Rouché-Capelli}), + \item $A\vec{x} = B$, se la ammette, ha un'unica soluzione + se e solo se $\Ker A = \zerovecset \iff \rg A = n$. + \end{itemize} + + Si definiscono tre operazioni sulle righe di una matrice $A$: + + \begin{enumerate} + \item l'operazione di scambio di riga, + \item l'operazione di moltiplicazione di una riga + per uno scalare non nullo, + \item la somma di un multiplo non nullo di una riga + ad un'altra riga distinta. + \end{enumerate} + + Queste operazioni non variano né $\Ker A$ né $\rg (A)$. Si possono effettuare le stesse medesime operazioni + sulle colonne (variando tuttavia $\Ker A$, ma lasciando + invariato $\Im A$ -- e quindi $\rg (A)$). L'algoritmo di eliminazione di Gauss + procede nel seguente modo: + + \begin{enumerate} + \item se $A$ ha una riga, l'algoritmo termina; + \item altrimenti si prenda la prima riga di $A$ con il primo elemento + non nullo e la si scambi con la prima riga di $A$ (in caso + non esista, si proceda all'ultimo passo), + \item per ogni riga di $A$ con primo elemento non nullo, + esclusa la prima, si sottragga un multiplo della prima riga in modo + tale che la riga risultante abbia il primo elemento nullo, + \item si ripeta l'algoritmo considerando come matrice $A$ la + matrice risultante dall'algoritmo senza la prima riga e la + prima colonna (in caso tale matrice non possa esistere, + l'algoritmo termina). + \end{enumerate} + + Si definiscono \textit{pivot} di una matrice l'insieme dei primi + elementi non nulli di ogni riga della matrice. + Il rango della matrice iniziale $A$ è pari al numero di \textit{pivot} + della matrice risultante dall'algoritmo di eliminazione di Gauss. + Una matrice che processata dall'algoritmo di eliminazione di Gauss + restituisce sé stessa è detta matrice a scala. + + Agendo solo attraverso + operazioni per riga, l'algoritmo di eliminazione di Gauss non + modifica $\Ker A$ (si può tuttavia integrare l'algoritmo con le + operazioni per colonna, perdendo quest'ultimo beneficio). + + Agendo + su una matrice a scala con operazioni per riga considerando + la matrice riflessa (ossia dove l'elemento $(1, 1)$ e $(m, n)$ sono + scambiati), si può ottenere una matrice a scala ridotta, + ossia un matrice dove tutti i pivot sono $1$ e dove tutti + gli elementi sulle colonne dei pivot, eccetto i pivot stessi, + sono nulli. + + Si definisce: + + \[I^{m \times n}_r = + \begin{pmatrix} + I_r + & \rvline & \bigzero \\ + \hline + \bigzero & \rvline & + \bigzero + \end{pmatrix} \in M(m, n, \KK). \] + + Per ogni applicazione lineare $f : V \to W$, con $\dim V = n$ e + $\dim W = m$ esistono due basi $\basis_V$, $\basis_W$ rispettivamente + di $V$ e $W$ tale che $M^{\basis_V}_{\basis_W}(f) = I^{m \times n}_r$, + dove $r=\rg(f)$ (è sufficiente completare con $I$ a base di $V$ una base + di $\Ker f$ e poi prendere come base di $W$ il completamento di $f(I)$ + su una base di $W$). + + Si definisce SD-equivalenza la relazione d'equivalenza su + $M(m, n, \KK)$ indotta dalla relazione $A \sim_{SD} B \iff \exists P \in + \GL(m, \KK)$, $Q \in \GL(n, \KK) \mid A=PBQ$. L'invariante completo + della SD-equivalenza è il rango: $\rg(A) = \rg(B) \iff A \sim_{SD} B$ + (infatti $\rg(A) = r \iff A \sim_{SD} I^{m \times n}_r$ -- è sufficiente + applicare il cambio di base e sfruttare il fatto che esistono + sicuramente due basi per cui $f_A$ ha $I^{m \times n}_r$ come + matrice associata). + + Poiché $I^{m \times n}_r$ ha sempre rango $r$, l'insieme + quoziente della SD-equivalenza su $M(m, n, \KK)$ è il seguente: + + \[ M(m, n, \KK)/{\sim_{SD}} = \left\{[\vec{0}], \left[I^{m \times n}_1\right], \ldots, \left[I^{m \times n}_{\min\{m, n\}}\right] \right\}, \] + + contenente esattamente $\min\{m, n\}$ elementi. L'unico elemento + di $[\vec{0}]$ è $\vec{0}$ stesso. + + \subsubsection{La regola di Cramer} + + Qualora $m=n$ e $A$ fosse invertibile (i.e. $\det(A) \neq 0$), + per calcolare il valore di $\vec{x}$ si può applicare + la regola di Cramer. + + Si definisce: + + \[ A_i^* = \begin{pmatrix}[c|c|c|c|c] + A^1 & \cdots & A^i \to B & \cdots & A^n + \end{pmatrix}, \] + + dove si sostituisce alla $i$-esima colonna di $A$ il vettore $B$. Allora + vale la seguente relazione: + + \[ \vec{x} = \frac{1}{\det(A)} \begin{pmatrix} + \det(A_1^*) \\ \vdots \\ \det(A_n^*) + \end{pmatrix}. \] + + \subsection{L'inverso (generalizzato e non) di una matrice} + + Si definisce matrice dei cofattori di una matrice $A \in M(n, \KK)$ la + seguente matrice: + + \[ \Cof A = \begin{pmatrix} + \Cof_{1,1}(A) & \ldots & \Cof_{1,n}(A) \\ + \vdots & \ddots & \vdots \\ + \Cof_{n,1}(A) & \ldots & \Cof_{n,n}(A), + \end{pmatrix}, \] + + dove, detta $A_{i,j}$ il minore di $A$ ottenuto eliminando + la $i$-esima riga e la $j$-esima colonna, si definisce il cofattore (o + complemento algebrico) nel seguente modo: + + \[ \Cof_{i,j}(A) = (-1)^{i+j} \det( A_{i, j}). \] + + Si definisce inoltre l'aggiunta classica: + + \[ \adj(A) = (\Cof A)^\top. \] + + Allora, se $A$ ammette un inverso (i.e. se $\det(A) \neq 0$), + vale la seguente relazione: + + \[ A^{-1} = \frac{1}{\det(A)} \adj(A). \] + + \vskip 0.05in + + Quindi, per esempio, $A^{-1}$ è a coefficienti + interi $\iff \det(A) = \pm 1$. + + Siano $A$, $B \in M(n, \KK)$. + + \begin{itemize} + \item $\adj(AB) = \adj(B)\adj(A)$, + \item $\adj(A^\top) = \adj(A)^\top$. + \end{itemize} + + Si definisce inverso generalizzato di una matrice $A \in M(m, n, \KK)$ + una matrice $X \in M(n, m, \KK) \mid AXA=A$. Ogni matrice ammette + un inverso generalizzato (è sufficiente considerare gli inversi + generalizzati di $I^{m \times n}_r$ e la SD-equivalenza di $A$ + con $I^{m \times n}_r$, dove $\rg(A)=r$). Se $m=n$ ed $A$ è invertibile, allora + $A^{-1}$ è l'unico inverso generalizzato di $A$. Gli inversi + generalizzati di $I^{m \times n}_r$ sono della forma: + + \[X = + \begin{pmatrix} + I_r + & \rvline & B \\ + \hline + C & \rvline & + D + \end{pmatrix} \in M(m, n, \KK). \] + + \subsection{Endomorfismi e similitudine} + + Si definisce la similitudine tra matrici su $M(n, \KK)$ come la relazione + di equivalenza determinata da $A \sim B \iff \exists P \in \GL(n, \KK) + \mid A = PBP^{-1}$. $A \sim B \implies \rg(A)=\rg(B)$, $\tr(A)=\tr(B)$, + $\det(A)=\det(B)$, $P_\lambda(A) = P_\lambda(B)$ (invarianti \textit{non completi} della similitudine). + Vale inoltre che $A \sim B \iff A$ e $B$ hanno la stessa forma + canonica di Jordan, a meno di permutazioni dei blocchi di Jordan + (invariante \textit{completo} della similitudine). La matrice + identità è l'unica matrice identica a sé stessa. + + Sia $p \in \End(V)$. Si dice che un endomorfismo è un automorfismo + se è un isomorfismo. Siano $\basis$, $\basis'$ due qualsiasi + basi di $V$. + + \begin{itemize} + \item $p$ automorfismo $\iff p$ iniettivo $\iff p$ surgettivo (è + sufficiente applicare la formula delle dimensioni), + \item $M^\basis_{\basis'}(id_V) M^{\basis'}_\basis(id_V) + = I_n$ (dunque entrambe le matrici sono invertibili e sono + l'una l'inverso dell'altra), + \item se $p$ è un automorfismo, $M^\basis_{\basis'}(p^{-1}) = + M^{\basis'}_\basis(p)^{-1}$, + \item $M^\basis_{\basis}(p) = \underbrace{M^{\basis'}_\basis (id_V)}_{P} \, + M^{\basis'}_{\basis'}(p) \, + \underbrace{M^{\basis}_{\basis'} (id_V)}_{P^{-1}}$ (ossia + $M^\basis_{\basis}(p) \sim M^{\basis'}_{\basis'}(p)$). + \end{itemize} + + $M^\basis_{\basis'}(id_V) M^{\basis'}_\basis(id_V) + = I_n$. Dunque entrambe le matrici sono invertibili. Inoltre + $M^\basis_\basis(id_V) = I_n$. + + \subsubsection{Duale, biduale e annullatore} + + Si definisce duale di uno spazio vettoriale $V$ lo + spazio $\dual{V} = \mathcal{L}(V, \KK)$, i cui elementi + sono detti funzionali. Analogamente + il biduale è il duale del duale di $V$: $\bidual{V} = \dual{(\dual{V})} = \mathcal{L}(\dual{V}, \KK)$. + + Sia data una base $\basis = \{\vec{v_1}, \ldots, \vec{v_n}\}$ di + uno spazio vettoriale $V$ di dimensione $n$. Allora $\dim \dual{V} + = \dim \mathcal{L}(V, \KK) = \dim V \cdot \dim \KK = \dim V$. Si definisce + il funzionale $\dual{\vec{v_i}}$ come l'applicazione lineare + univocamente determinata dalla relazione: + + \[ \dual{\vec{v_i}}(\vec{v_j}) = \delta_{ij}. \] + + \vskip 0.05in + + Sia $\basis^* = \{\vec{v_1}^*, \ldots, \vec{v_n}^*\}$. Allora + $\basis^*$ è una base di $\dual{V}$. Poiché $V$ e $\dual{V}$ + hanno la stesso dimensione, tali spazi sono isomorfi, sebbene + non canonicamente. Ciononostante, $V$ e $\bidual{V}$ sono + canonicamente isomorfi tramite l'isomorfismo: + + \[ \bidual{\varphi} : V \to \bidual{V}, \; \vec{v} \mapsto \restr{\val}{\dual{V}}, \] + + che associa ad ogni vettore $\vec{v}$ la funzione + di valutazione in una funzionale in $\vec{v}$, ossia: + + \[ \restr{\val}{\dual{V}} : \dual{V} \to \KK, \; f \mapsto f(\vec{v}). \] + + Sia $U \subseteq V$ un sottospazio di $V$. + Si definisce il sottospazio di $\mathcal{L}(V, W)$: + + \[ \Ann_{\mathcal{L}(V, W)}(U) = \left\{ f \in \mathcal{L}(V, W) \mid f(U) = \zerovecset \right\}. \] + + Se $V$ è a dimensione finita, la dimensione di + $\Ann(U)$ è pari a $(\dim V - \dim U) \cdot \dim W$ (è sufficiente + prendere una base di $U$, completarla a base di $V$ e + notare che $f(U) = \zerovecset \iff$ ogni valutazione + in $f$ degli elementi della base di $U$ è nullo $\iff$ la matrice + associata di $f$ ha tutte colonne nulle in corrispondenza degli + elementi della base di $U$). + + Si scrive semplicemente $\Ann(U)$ quando $W=\KK$ (ossia + quando le funzioni sono funzionali di $V$). In tal + caso $\dim \Ann(U) = \dim V - \dim U$. + + \begin{itemize} + \item $\bidual{\varphi}(U) \subseteq \Ann(\Ann(U))$, + \item se $V$ è a dimensione finita, $\bidual{\varphi}(U) = \bidual{U} = \Ann(\Ann(U))$ (è sufficiente + applicare la formula delle dimensioni $\restr{\bidual{\varphi}}{U}$ e notare l'uguaglianza + tra le due dimensioni), + \item se $V$ è a dimensione finita e $W$ è un altro + sottospazio di $V$, + $U = W \iff \Ann(U) = \Ann(W)$ (è sufficiente + considerare $\Ann(\Ann(U)) = \Ann(\Ann(W))$ e + applicare la proposizione precedente, ricordandosi + che $\bidual{\varphi}$ è un isomorfismo, ed è + dunque iniettivo). + \end{itemize} + + Si definisce l'applicazione trasposta $^\top$ da $\mathcal{L}(V, W)$ a + $\mathcal{L}(\dual{W}, \dual{V})$ in modo tale che $f^\top(g) + = g \circ f \in \dual{V}$. Siano $f$, $g \in \mathcal{L}(V,W)$ e + sia $h \in \mathcal{L}(W,Z)$. + + \begin{itemize} + \item $(f+g)^\top = f^\top + g^\top$, + \item $(\lambda f)^\top = \lambda f^\top$, + \item se $f$ è invertibile, $(f^{-1})^\top = (f^\top)^{-1}$, + \item $(h \circ f)^\top = f^\top \circ h^\top$. + \end{itemize} + + Siano $\basis_V$, $\basis_W$ due basi rispettivamente di $V$ e + di $W$. Allora vale la seguente relazione: + + \[ M^{\basis_W^*}_{\basis_V^*}(f^\top) = M^{\basis_V}_{\basis_W}(f)^\top. \] + + \subsection{Applicazioni multilineari} + + Sia $f : V_1 \times \ldots \times V_n \to W$ un'applicazione, dove + $V_i$ è uno spazio vettoriale $\forall i \leq n$, così come $W$. Tale + applicazione si dice $n$-lineare ed appartiene allo spazio + $\Mult(V_1 \times \ldots \times V_n, W)$, se è lineare in ogni sua coordinata, ossia se: + + \begin{itemize} + \item $f(x_1, \ldots, x_i + y_i, \ldots, x_n) = + f(x_1, \ldots, x_i, \ldots, x_n) + f(x_1, \ldots, y_i, \ldots, x_n)$, + \item $f(x_1, \ldots, \alpha x_i, \ldots, x_n) = \alpha f(x_1, \ldots, x_i, \ldots, x_n)$. + \end{itemize} + + Sia $W=\KK$, e siano tutti gli spazi $V_i$ fondati su tale campo: allora + $\Mult(V_1 \times \ldots \times V_n, \KK)$ si scrive anche come $V_1^* \otimes \ldots \otimes V_n^*$, e tale spazio + è detto prodotto tensoriale tra $V_1$, ..., $V_n$. + Sia $V_i$ di dimensione finita $\forall i \leq n$. Siano $\basis_{V_i} = \left\{ \vec{v^{(i)}_1}, \ldots, \vec{v^{(i)}_{k_i}} \right\}$ base + di $V_i$, dove $k_i = \dim V_i$. + Si definisce l'applicazione $n$-lineare $\dual{\vec{v^{(1)}_{j_1}}} + \otimes \cdots \otimes \dual{\vec{v^{(n)}_{j_n}}}\in \Mult(V_1 \times \ldots + \times V_n, \KK)$ univocamente determinata dalla relazione: + + \[ \dual{\vec{v^{(1)}_{j_1}}} \otimes \cdots \otimes \dual{\vec{v^{(n)}_{j_n}}}(\vec{w_1}, \ldots, \vec{w_n}) = \dual{\vec{v^{(1)}_{j_1}}}(\vec{w_1}) \cdots \dual{\vec{v^{(n)}_{j_n}}}(\vec{w_n}). \] + + Si definisce l'insieme $\basis_{\otimes}$ nel seguente modo: + + \[ \basis_{\otimes} = \left\{ \dual{\vec{v^{(1)}_{j_1}}} \otimes \cdots \otimes \dual{\vec{v^{(n)}_{j_n}}} \mid 1 \leq j_1 \leq k_1, \, \ldots, \, 1 \leq j_n \leq k_n \right\}. \] + + Poiché ogni applicazione $n$-lineare è univocamente determinata + dai valori che assume ogni combinazione degli elementi delle basi + degli spazi $V_i$, vi è un isomorfismo tra $\Mult(V_1 \times \ldots + \times V_n, \KK)$ e $\KK^{\basis_{V_1} \times \cdots \times \basis_{V_n}}$, che ha dimensione $\prod_{i=1}^n k_i = k$. Pertanto + anche $\dim \Mult(V_1 \times \ldots \times V_n, \KK) = k$. + + Poiché $\basis_{\otimes}$ genera $\Mult(V_1 \times \ldots + \times V_n, \KK)$ e i suoi elementi sono tanti quanto è la + dimensione dello spazio, tale insieme è una base di $\Mult(V_1 \times + \ldots \times V_n, \KK)$. + + Se $V_i = V_1 = V$ $\forall i \leq n$, si dice che $\Mult(V^n, \KK)$ + è lo spazio delle forme $n$-lineari di $V$. + + \subsubsection{Applicazioni multilineari simmetriche} + + Sia $V$ uno spazio di dimensione $n$. Una + forma $k$-lineare $f$ si dice simmetrica + ed appartiene allo spazio $\Sym^k(V)$ se: + + \[ f(\vec{x_1}, \ldots, \vec{x_k}) = f(\vec{x_{\sigma(1)}}, \ldots, \vec{x_{\sigma(k)}}), \quad \forall \sigma \in S_k. \] + + Poiché ogni applicazione $n$-lineare simmetrica è univocamente + determinata dai valori che assume negli elementi della base + disposti in modo non decrescente, $\dim \Sym^k(V) = \binom{n+k-1}{k}$. + + Sia $\basis = \{\vec{v_1}, \ldots, \vec{v_n}\}$ una base + di $V$. Dato un insieme di indici non decrescente $I$, + si definisce il prodotto simmetrico (o \textit{prodotto vee}) + $\dual{\vec{v_{i_1}}} \vee \cdots \vee \dual{\vec{v_{i_k}}}$ + tra elementi della base come la forma $k$-lineare simmetrica + determinata dalla relazione: + + \[ \dual{\vec{v_{i_1}}} \vee \cdots \vee \dual{\vec{v_{i_k}}} = \sum_{\sigma \in S_k} \dual{\vec{v_{i_{\sigma(1)}}}} \otimes \cdots \otimes \dual{\vec{v_{i_{\sigma(k)}}}}. \] + + Si definisce l'insieme: + + \[\basis_{\Sym} = \left\{ \dual{\vec{v_{i_1}}} \vee \cdots \vee \dual{\vec{v_{i_k}}} \mid 1 \leq i_1 \leq \cdots \leq i_k \leq n \right\}. \] + + L'insieme $\basis_{\Sym}$ è sia generatore che linearmente + indipendente su $\Sym^k(V)$, ed è dunque base. Allora + $\dim \Sym^k(V) = \binom{n+k-1}{k}$. + + \subsubsection{Applicazioni multilineari alternanti} + + Sia $V$ uno spazio di dimensione $n$. Una forma + $k$-lineare $f$ si dice alternante (o antisimmetrica) + ed appartiene allo spazio $\Lambda^k(V)$ (talvolta scritto + come $\operatorname{Alt}^k(V)$) se: + + \[ f(x_1, \ldots, x_k) = 0 \impliedby \exists \, i, j \leq k \mid x_i = x_j. \] + + \vskip 0.05in + + Questo implica che: + + \[ f(x_1, \ldots, x_k) = \sgn(\sigma) f(x_{\sigma(1)}, \ldots, x_{\sigma(n)}), \quad \forall \sigma \in S_k \] + + Se $k > n$, un argomento della base di $V$ si ripete sempre nel + computo $f$ negli elementi della base, e quindi ogni alternante è + pari a $\vec{0}$, ossia $\dim \Lambda^k(V) = 0$. + + Sia $\basis = \{\vec{v_1}, \ldots, \vec{v_n}\}$ una base + di $V$. Dato un insieme di indici crescente $I$, + si definisce il prodotto esterno (o \textit{prodotto wedge}) + $\dual{\vec{v_{i_1}}} \wedge \cdots \wedge \dual{\vec{v_{i_k}}}$ + tra elementi della base come la forma $k$-lineare alternante + determinata dalla relazione: + + \[ \dual{\vec{v_{i_1}}} \wedge \cdots \wedge \dual{\vec{v_{i_k}}} = \sum_{\sigma \in S_k} \sgn(\sigma) \, \dual{\vec{v_{i_{\sigma(1)}}}} \otimes \cdots \otimes \dual{\vec{v_{i_{\sigma(k)}}}}. \] + + Si definisce l'insieme: + + \[\basis_{\Lambda} = \left\{ \dual{\vec{v_{i_1}}} \wedge \cdots \wedge \dual{\vec{v_{i_k}}} \mid 1 \leq i_1 < \cdots < i_k \leq n \right\}. \] + + L'insieme $\basis_{\Lambda}$ è sia generatore che linearmente + indipendente su $\Lambda^k(V)$, ed è dunque base. Allora + $\dim \Lambda^k(V) = \binom{n}{k}$. Riassumendo si può scrivere: + + \[\dim \Lambda^k(V) = \begin{cases} 0 & \text{se } k > n\,, \\ \binom{n}{k} & \text{altrimenti}. \end{cases}\] + + Quindi è quasi sempre vero che: + + \[ \underbrace{\dim \Sym^k(V)}_{= \, \binom{n+k-1}{k}} + \underbrace{\dim \Lambda^k(V)}_{\leq \, \binom{n}{k}} < \underbrace{\dim \Mult(V^k, \KK)}_{=\,n^k}, \] + + e dunque che $\Sym^k(V) + \Lambda^k(V) \neq \Mult(V^k, \KK)$. + + + + \subsection{Determinante di una matrice} + + Si definisce il determinante $\det$ di una matrice di taglia + $n \times n$ come l'unica forma $n$-lineare alternante di $(\KK^n)^n$ + tale che $\det(\vec{e_1}, \ldots, \vec{e_n}) = 1$ (infatti + $\dim \Lambda^n (V) = \binom{n}{n} = 1$, e quindi ogni forma + alternante è multipla delle altre, eccetto per lo zero). + + Equivalentemente $\det = \dual{\vec{e_1}} \, \wedge \cdots \wedge \, \dual{\vec{e_n}}$. + + Siano $A$, $B \in M(n, \KK)$. Si scrive + $\det(A)$ per indicare $\det(A_1, \ldots, A_n)$. Vale pertanto la + seguente relazione: + + \[ \det(A) = \sum_{\sigma \in S_n} \sgn(\sigma) \, a_{1\sigma(1)} \cdots a_{n\sigma(n)}. \] + + \begin{itemize} + \item $\det(I_n) = 1$, + \item $\det \begin{pmatrix} + a & b \\ c & d + \end{pmatrix} = ad-bc$, + \item $\det \begin{pmatrix} + a & b & c \\ d & e & f \\ g & h & i + \end{pmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg)$, + \item $\det(A) \neq 0 \iff A$ invertibile (ossia non singolare), + \item $\det(\lambda A) = \lambda^n A$, + \item $\det(A) = \det(A^\top)$ (è sufficiente applicare la definizione + di $\det$ e manipolare algebricamente il risultato per evidenziare + l'uguaglianza), + \item se $A$ è antisimmetrica, $n$ è dispari e $\Char \KK \neq 2$, + $\det(A) = \det(-A^\top) = (-1)^n \det(A^\top) = (-1)^n \det(A) = -\det(A) \implies \det(A) = 0$ (quindi ogni matrice antisimmetrica di taglia + dispari non è invertibile), + \item $\det(AB) = \det(A)\det(B)$ (\textit{teorema di Binet} -- è + sufficiente considerare la forma $\frac{\det(AB)}{\det(B)}$ in + funzione delle righe di $A$ e determinare che tale forma + è alternante e che vale $1$ nell'identità, e che, per l'unicità + del determinante, deve obbligatoriamente essere pari a + $\det(A)$), + \item se $A$ è invertibile, $\det(A^{-1}) = \det(A)^{-1}$, + \item $\det \begin{pmatrix} + \lambda_{1} & & \\ + & \ddots & \\ + & & \lambda_{n} + \end{pmatrix} = \det(\lambda_1 \vec{e_1}, \ldots, \lambda_n \vec{e_n}) = \prod_{i=1}^n \lambda_i$, + \item se $A$ è triangolare superiore (o inferiore), allora $\det(A)$ è + il prodotto degli elementi sulla sua diagonale principale, + \item $\det(A_1, \ldots, A_n) = \sgn(\sigma) \det(A_{\sigma(1)}, \ldots, A_{\sigma(n)})$, $\forall \sigma \in S_n$ (infatti $\det$ è alternante), + \item $\det \begin{pmatrix} + A + & \rvline & B \\ + \hline + C & \rvline & + D + \end{pmatrix} = \det(AD-BC)$, se $C$ e $D$ commutano e $D$ è invertibile, + \item $\det \begin{pmatrix} + A + & \rvline & B \\ + \hline + 0 & \rvline & + C + \end{pmatrix} = \det(A)\det(C)$, + \item se $A$ è nilpotente (ossia se $\exists k \mid A^k = 0$), + $\det(A) = 0$, + \item se $A$ è idempotente (ossia se $A^2 = A$), allora + $\det(A) = 1$ o $\det(A) = 0$, + \item se $A$ è ortogonale (ossia se $AA^\top = I_n$), allora + $\det(A) = \pm 1$, + \item se $A$ è un'involuzione (ossia se $A^2 = I_n$), allora + $\det(A) = \pm 1$, + \end{itemize} + + Le operazioni del terzo tipo dell'algoritmo di eliminazione + di Gauss (ossia l'aggiunta a una riga di un multiplo di un'altra + riga -- a patto che le due righe siano distinte) non alterano il + determinante della matrice iniziale, mentre lo scambio di righe + ne inverte il segno (corrisponde a una trasposizione di $S_n$). + L'operazione del secondo tipo (la moltiplicazione di una riga + per uno scalare) altera il determinante moltiplicandolo per + tale scalare. + + Inoltre, se $D$ è invertibile, vale la seguente scomposizione: + + \[ \begin{pmatrix} + A + & \rvline & B \\ + \hline + C & \rvline & + D + \end{pmatrix} = \begin{pmatrix} + I_k + & \rvline & BD^{-1} \\ + \hline + 0 & \rvline & + I_k + \end{pmatrix} + \begin{pmatrix} + A-BD^{-1}C + & \rvline & 0 \\ + \hline + 0 & \rvline & + D + \end{pmatrix} + \begin{pmatrix} + I_k + & \rvline & 0 \\ + \hline + D^{-1}C & \rvline & + I_k + \end{pmatrix}, \] + + dove $k \times k$ è la taglia di $A$. Pertanto vale + la seguente relazione, sempre se $D$ è invertibile: + + \[ \det \begin{pmatrix} + A + & \rvline & B \\ + \hline + C & \rvline & + D + \end{pmatrix} = \det(A-BD^{-1}C)\det(D). \] + + È possibile computare il determinante di $A$, scelta la riga $i$, mediante lo + sviluppo di Laplace: + + \[ \det(A) = \sum_{j=1}^n a_{ij} \Cof_{i,j}(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} \det(A_{i,j}). \] + + Si definisce matrice di Vandermonde una matrice $A \in M(n, \KK)$ della + forma: + + \[ A = \begin{pmatrix} + 1 & x_1 & x_1^2 & \dots & x_1^{n-1}\\ + 1 & x_2 & x_2^2 & \dots & x_2^{n-1}\\ + \vdots & \vdots & \vdots & \ddots &\vdots \\ + 1 & x_n & x_n^2 & \dots & x_n^{n-1}. + \end{pmatrix} \] + + Vale allora che: + + \[ \det(A) = \prod_{1 \leq i < j \leq n} (x_j - x_i), \] + + verificabile notando che $\det(A)$ è di grado $\frac{n(n-1)}{2}$ e + che ponendo $x_i = x_j$ per una coppia $(i, j)$, tale matrice + ha due righe uguali, e quindi determinante nullo $\implies (x_j - x_i) \mid \det(A) \overbrace{\implies}^{\text{UFD}} \det(A) = \prod_{1 \leq i < j \leq n} (x_j - x_i) $. + + Pertanto una matrice di Vandermonde è invertibile se e solo se la sua + seconda colonna contiene tutti scalari distinti nelle coordinate. Tale + matrice risulta utile nello studio dell'interpolazione di Lagrange + (ossia nella dimostrazione dell'unicità del polinomio di $n-1$ grado + tale che $p(\alpha_i) = \beta_i$ per $i$ coppie ($\alpha_i$, $\beta_i$) con + $\alpha_i$ tutti distinti). + + \subsection{Autovalori e diagonalizzabilità} + + Sia $f \in \End(V)$. Si dice che $\lambda \in \KK$ è un autovalore + di $f$ se e solo se $\exists \vec{v} \neq \vec{0}$, $\vec{v} \in V$ + 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 + per qualsiasi cambio di base applicato ad una matrice associata + 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 + $V_\lambda = \Ker(f - \lambda \Idv)$ lo spazio degli autovettori + relativo a $\lambda$, detto autospazio di $\lambda$. + + Si definisce la molteplicità algebrica $\mu_{a,f}(\lambda)$ di un autovalore + $\lambda$ come la molteplicità che assume come radice del polinomio + $p_f(\lambda)$. Si definisce la molteplicità geometrica + $\mu_{g,f}(\lambda)$ di un autovalore $\lambda$ come la dimensione + del suo autospazio $V_\lambda$. Quando è noto l'endomorfismo + che si sta considerando si omette la dicitura $f$ nel pedice delle + molteplicità. + + \begin{itemize} + \item $p_f(\lambda)$ ha sempre grado $n = \dim V$, + \item $p_f(\lambda)$ è sempre monico a meno del segno, + \item il coefficiente di $\lambda^n$ è sempre $(-1)^n$, + \item il coefficiente di $\lambda^{n-1}$ è $(-1)^{n+1} \tr(f)$, + \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 + $n$ 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 + 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 autovettori relativi ad autovalori distinti sono sempre + linearmente indipendenti, + \item dati $\lambda_1$, ..., $\lambda_k$ autovalori di $f$, + gli spazi $V_{\lambda_1}$, ..., $V_{\lambda_k}$ sono sempre + in somma diretta, + \item $\sum_{i=1}^k \mu_a(\lambda_i)$ corrisponde al numero + di fattori lineari di $p_f(\lambda)$, + \item $\sum_{i=1}^k \mu_a(\lambda_i) = n \iff$ $p_f(\lambda)$ + è completamente fattorizzabile in $\KK[\lambda]$, + \item vale sempre la disuguaglianza $n \geq \mu_a(\lambda) \geq + \mu_g(\lambda) \geq 1$ (è sufficiente considerare una + base di $V_\lambda$ estesa a base di $V$ e calcolarne il + polinomio caratteristico sfruttando i blocchi della matrice + associata, notando che $\mu_g(\lambda)$ deve forzatamente essere + 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 + 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), + \item se $W \subseteq V$ è un sottospazio $f$-invariante, + ed estesa una base $\basis_W$ di $W$ ad una $\basis$ di $V$, + detto $U = \Span(\basis \setminus \basis_W)$ il supplementare di $W$ che si ottiene da tale base $\basis$, vale + che $\charpoly{f} = \charpolyrestr{f}{W} \cdot \charpoly{\hat{f}}$, + dove $\hat{f} : V/W \to V/W$ è tale che $\hat{f}(\vec{u} + W) = f(\vec{u}) + W$ (come prima, è sufficiente considerare una matrice + a blocchi), + \item se $V = W \oplus U$, dove sia $W$ che $U$ sono $f$-invarianti, + allora $\charpoly{f} = \charpolyrestr{f}{W} \cdot \charpolyrestr{f}{U}$ (la matrice associata in un'unione di basi + di $W$ e $U$ è infatti diagonale a blocchi). + \end{itemize} + + Si dice che $f$ è diagonalizzabile se $V$ ammette una base per cui + la matrice associata di $f$ è diagonale, o equivalentemente se, + dati $\lambda_1$, ..., $\lambda_k$ autovalori di $f$, si verifica + che: + + \[ V = V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k}. \] + + Ancora in modo equivalente si può dire che $f$ è diagonalizzabile + se e solo se: + + \[ \begin{cases} \sum_{i=1}^k \mu_a(\lambda_i) = n, \\ \mu_g(\lambda_i) = \mu_a(\lambda_i) \; \forall 1 \leq i \leq k, \end{cases} \] + + ossia se il polinomio caratteristico è completamente fattorizzabile + in $\KK[\lambda]$ (se non lo fosse, la somma diretta + $V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k}$ avrebbe + forzatamente dimensione minore di $V$, ed esisterebbero altri + autovalori in un qualsiasi campo di spezzamento di $p_f(\lambda)$) e se $\sum_{i=1}^k \mu_g(\lambda_i) = n$. Tale condizione, in un + campo algebricamente chiuso, si riduce a $\mu_g(\lambda_i) = \mu_a(\lambda_i)$, $\forall 1 \leq i \leq k$. + + Considerando la forma canonica di Jordan di $f$, si osserva anche + 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. + + Data $f$ diagonalizzabile, la matrice diagonale $J$ a cui $f$ è + associata è, dati gli autovalori $\lambda_1$, ..., $\lambda_k$, + una matrice diagonale dove $\lambda_i$ compare sulla diagonale + esattamente $\mu_g(\lambda_i)$ volte. + + Data $A \in M(n, \KK)$, $A$ è diagonalizzabile se e solo se $f_A$, + l'applicazione indotta dalla matrice $A$, è diagonalizzabile, + ossia se $A$ è simile ad una matrice diagonale $J$, computabile + come prima. Si scrive in particolare $p_A(\lambda)$ per indicare + $p_{f_A}(\lambda)$. + + Una matrice $P \in \GL(M(n, \KK))$ + tale che $A = P J P\inv$, è tale che $AP = PJ$: presa la $i$-esima + colonna, allora, $AP^{(i)} = PJ^{(i)} = P^{(i)}$; ossia è sufficiente + costruire una matrice $P$ dove l'$i$-esima colonna è un autovettore + relativo all'autovalore presente in $J_{ii}$ linearmente indipendente + con gli altri autovettori presenti in $P$ relativi allo stesso + autovalore (esattamente nello stesso modo in cui si costruisce in + generale tale $P$ con la forma canonica di Jordan). + + Se $A$ e $B$ sono diagonalizzabili, allora $A \sim B \iff p_A(\lambda) = + p_B(\lambda)$ (infatti due matrici diagonali hanno lo stesso polinomio + caratteristico se e solo se compaiono gli stessi identici autovalori). + + 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 + $f$. + + 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 + di $g$ sono diagonali. Vale in particolare che $f$ e $g$ sono + simultaneamente diagonalizzabili se e solo se $f \circ g = g \circ f$. + Per trovare tale base è sufficiente, dati $\lambda_1$, ..., + $\lambda_k$ autovalori di $f$, considerare $\restr{g}{V_{\lambda_i}}$ + $\forall 1 \leq i \leq k$ ($V_{\lambda_i}$ è infatti $g$-invariante, + dacché, per $\vec{v} \in V_{\lambda_i}$, $f(g(\vec{v})) = + g(f(\vec{v})) = g(\lambda_i \vec{v}) = \lambda_i g(\vec{v}) \implies + g(\vec{v}) \in V_{\lambda_i}$), che, essendo una restrizione di + un endomorfismo diagonalizzabile su un sottospazio invariante, è diagonalizzabile: presa allora + una base di autovettori di $\restr{g}{V_{\lambda_i}}$, questi sono + 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. + + %\item vale sempre che $p_f(f) = 0$ (teorema di Hamilton-Cayley -- + % data una matrice associata $A$ di $f$, è sufficiente studiare + % l'identità + % $(A-\lambda I_n) \cdot \adj(A-\lambda I_n) = p_f(\lambda) I_n$) + + \vfill + \hrule + ~\\ + Gabriel Antonio Videtta, \url{https://poisson.phc.dm.unipi.it/~videtta/} + \end{multicols} + +\end{document} \ No newline at end of file diff --git a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-24/main.pdf b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-24, Es. forma canonica di Jordan e autospazi generalizzati/main.pdf similarity index 99% rename from Geometria I/Teoria spettrale degli endomorfismi/2023-03-24/main.pdf rename to Geometria I/Teoria spettrale degli endomorfismi/2023-03-24, Es. forma canonica di Jordan e autospazi generalizzati/main.pdf index 511d6ba05be2ade9a024731f4f73be420983f75c..07c2cd07c7f4f23a5ff54fe04a13499371ad16cb 100644 GIT binary patch delta 120 zcmbPpnS0JXCo&|12YRp oV>c&rQ)5dvOBW+&6DKD_XJZ$06C-mI7dr(TLQ1CBzhsgH0P%(*egFUf delta 120 zcmbPpnS0JXGcSGa|3fH o0}DqNR}&K>Lla{s3r9y2S3?s+6GsDA13Lv9LQ1CBzhsgH0N*4c4FCWD diff --git a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-24/main.tex b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-24, Es. forma canonica di Jordan e autospazi generalizzati/main.tex similarity index 100% rename from Geometria I/Teoria spettrale degli endomorfismi/2023-03-24/main.tex rename to Geometria I/Teoria spettrale degli endomorfismi/2023-03-24, Es. forma canonica di Jordan e autospazi generalizzati/main.tex diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index ba6d93e..32ac7ac 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -84,12 +84,29 @@ #1\arrowvert_{#2} } +\newcommand{\innprod}[2]{\langle {#1,#2} \rangle} + +\newcommand{\zerovecset}{\{\vec 0\}} +\newcommand{\bigzero}{\mbox{0}} +\newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} + +\newcommand{\Idv}{Id_V} +\DeclareMathOperator{\Bil}{Bil} +\DeclareMathOperator{\Mult}{Mult} +\DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\adj}{adj} +\DeclareMathOperator{\Cof}{Cof} + +\newcommand{\Eigsp}[1]{V_{\lambda}} +\newcommand{\Gensp}[1]{\widetilde{V_{\lambda}}} \newcommand{\eigsp}[1]{V_{\lambda_{#1}}} \newcommand{\gensp}[1]{\widetilde{V_{\lambda_{#1}}}} \DeclareMathOperator{\val}{val} \DeclareMathOperator{\Span}{Span} \newcommand{\charpoly}[1]{p_{#1}} +\newcommand{\charpolyrestr}[2]{p_{#1\arrowvert_#2}\hspace{-1pt}(\lambda)} \newcommand{\minpoly}[1]{\varphi_{#1}} \newcommand{\valf}{\val_f} \newcommand{\valfv}{\val_{f,\V}}