From 0f21dbc536d680420f8fdf90f1780f88ff1b0348 Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 20 Mar 2023 17:53:08 +0100 Subject: [PATCH] feat(geometria): aggiunge gli appunti dell'esercitazione del 20/03/2023 --- .../main.tex | 150 ++++++++++++++ .../personal_commands.sty | 189 ++++++++++++++++++ ...mi per la ricerca del polinomio minimo.pdf | Bin 0 -> 180007 bytes 3 files changed, 339 insertions(+) create mode 100644 Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex create mode 100644 Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/personal_commands.sty create mode 100644 Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo.pdf diff --git a/Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex b/Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex new file mode 100644 index 0000000..cab87dd --- /dev/null +++ b/Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex @@ -0,0 +1,150 @@ +\documentclass[11pt]{article} +\usepackage{personal_commands} +\usepackage[italian]{babel} + +\title{\textbf{Note del corso di Geometria 1}} +\author{Gabriel Antonio Videtta} +\date{\today} + +\begin{document} + +\maketitle + +\begin{center} + \Large \textbf{Esercitazione: algoritmi per la ricerca del polinomio minimo} +\end{center} + +\begin{definition} Dato $f \in \End(V)$, si definisce +come $\val_{f, \vec{v}}$ l'applicazione +lineare da $\KK[x]$ in $V$ tale che +$\val_{f, \vec{v}}(p) = p(f)(\vec{v})$. +\end{definition} + +\begin{remark} Vi sono varie proprietà +che legano $\Ker \val_{f, \vec{v}}$ +a $\Ker \val_f$, ed in particolare +il generatore monico di $\Ker \val_{f, \vec{v}}$ +$\varphi_{f, \vec{v}}$ a quello +$\varphi_{f}$ di $\Ker \val_f$, ossia al polinomio minimo di $f$. \\ + +\li $\varphi_{f, \vec{v}} \mid \varphi_{f}$, $\forall \vec{v} \in V$. \\ +\li $\varphi_{f} = \mcm(\varphi_{f, \vec{v_1}}, ..., \varphi_{f, \vec{v_n}}).$, dove i $\vec{v_1}$, ..., +$\vec{v_n}$ formano una base di $V$. \\ +\end{remark} + +\begin{example} Sia $A = \begin{pmatrix}2 & 0 & 0 \\ 1 & -1 & 3 \\ 1 & 3 & -1 \end{pmatrix}$. Allora +si possono considerare le seguenti +catene: \\ + +\li $\vec{e_1} \mapsto 2\vec{e_1} + \vec{e_2} + \vec{e_3} \mapsto 2(2\vec{e_1} + \vec{e_2} + \vec{e_3}) + (-\vec{e_2} + 3\vec{e_3}) ++ (3\vec{e_2} -\vec{e_3}) = 4\vec{e_1} + 4\vec{e_2} + 4\vec{e_3} = 4 A \vec{e_1} - 4 \vec{e_1}$. +Pertanto $A^2 \vec{e_1} - 4 A \vec{e_1} + 4 \vec{e_1} = \vec{0}$. +Essendo $A \vec{e_1}$ e $\vec{e_1}$ linearmente indipendenti, si conclude +che $\varphi_{A, \vec{e_1}}(x) = x^2 - 4x + 4 = (x-2)^2$. \\ +\li $\vec{e_2} \mapsto -\vec{e_2} + 3\vec{e_3} \mapsto -(-\vec{e_2} + 3\vec{e_3}) + 3(3\vec{e_2} -\vec{e_3}) = 10\vec{e_2} - 6\vec{e_3} = -2(-\vec{e_2} + 3\vec{e_3}) + 8\vec{e_2}$. +Si conclude dunque che $\varphi_{A, \vec{e_2}}(x) = x^2+2x-8 = (x-2)(x+4)$. \\ +\li $\vec{e_3} \mapsto 3\Vec{e_2}-\Vec{e_3} \mapsto 3(-\Vec{e_2} + 3\Vec{e_3}) - (3\Vec{e_2} - \Vec{e_3}) = -6\Vec{e_2} + 10\Vec{e_3} = -2(3\Vec{e_2} - \Vec{e_3}) + 8\Vec{e_3}$. Dunque +$\varphi_{A,\Vec{e_3}}(x) = x^2+2x-8 = \varphi_{A,\Vec{e_2}}(x)$. \\ + +Pertanto $\varphi_A(x) = \mcm(\varphi_{A,\Vec{e_1}}(x),\varphi_{A,\Vec{e_2}}(x),\varphi_{A,\Vec{e_3}}(x)) = (x-2)^2(x+4)$. +\end{example} + +\begin{definition} + Si dice che un vettore $\Vec{v}$ è \textit{ciclico} su $f$ se il ciclo + $\Span(\vec{v}, f(\vec{v}), f^2(\Vec{v}), ...)$ coincide + con $V$. +\end{definition} + +\begin{remark} + Riguardo all'esistenza di un vettore ciclico si possono + fare alcune osservazioni. \\ + + \li Se esiste un vettore $\vec{v}$ ciclico rispetto a $f$, i primi $n = \dim V$ + vettori del suo ciclo devono essere linearmente indipendenti + (altrimenti non potrebbe generare $V$), e quindi $\varphi_{f,\Vec{v}}$ deve avere grado $n$. Allora + anche $\varphi_f$ deve avere grado $n$, ossia lo stesso + grado di $p_f$. Allora, dal momento che $\varphi_f \mid p_f$ + e $\deg \varphi_f = \deg p_f$, deve valere necessariamente + $\varphi_f = \pm p_f$. \\ + \li Dal momento che $\varphi_{f,\Vec{v}}$ è monico, ha lo stesso grado + di $\varphi_f$ e lo divide, deve anche valere che $\varphi_{f,\Vec{v}} = \varphi_f$. \\ + \li Nella base ordinata $\basis$ costituita dai primi $n$ vettori del ciclo di $\Vec{v}$, la matrice associata di $f$ è della forma: + + \[ M_{\basis}(f) = \begin{pmatrix} + 0 & 0 & \ldots & \ldots & 0 & -a_0 \\ + 1 & 0 & \ldots & \ldots & 0 & -a_1 \\ + 0 & 1 & \ddots & & \vdots & \vdots \\ + \vdots & \vdots & & \ddots & 0 & -a_{n-2} \\ + 0 & 0 & 0 & \ldots & 1 & -a_{n-1} + \end{pmatrix}, \] + + dove gli $a_i$ sono i coefficienti di $\varphi_f(x) = \varphi_{f,\Vec{v}} = x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0$. +\end{remark} + +\begin{proposition} +Se $\KK$ è un campo infinito\footnote{In realtà la tesi è vera per qualsiasi campo, benché la dimostrazione che è stata fornita sia valida solo per campi infiniti.}, esiste sempre un vettore $\vec{v} \in V$ tale che $\varphi_{f, \vec{v}} = \varphi_f$. +\end{proposition} + +\begin{proof} +Si definisce il seguente insieme: + +\[ S = \{ \varphi_{f, \vec{v}} \mid \vec{v} \in V \}. \] + +\vskip 0.1in + +Poiché $S$ è un sottoinsieme dei divisori di $\phi_f$, $S$ è finito. +In particolare $\exists v_1$, ..., $v_n$ tali che $S = \{ \varphi_{f, \vec{v_1}}, ..., \varphi_{f, \vec{v_n}} \}$. Dal momento +che ogni $\vec{v} \in V$ è associato +ad un unico polinomio caratteristico, +vale che $V = \bigcup_{i=1}^n \Ker \varphi_{f, \vec{v_i}}$. Tuttavia, se +tutti i $\Ker \varphi_{f, \vec{v_i}}$ +fossero propri, questo sarebbe +impossibile, dal momento che uno spazio vettoriale fondato su un campo finito non può essere unione finita di sottospazi propri. Quindi $V = \Ker \varphi_{f, \vec{v_i}}$ per un $i$ tale che $1 \leq i \leq n$. Allora +$\varphi_f \mid \varphi_{f, \vec{v_i}}$, da cui si ricava +l'uguaglianza. +\end{proof} + +\begin{theorem} +Lo spazio $V$ ammette un vettore ciclico su $f \in \End(V)$ se e +solo se $p_f = \pm \varphi_f$. +\end{theorem} + +\begin{proof} Si dimostrano le due implicazioni separatamente. \\ + +($\implies$) Dall'osservazione precedente. \\ +($\impliedby$) Dalla proposizione precedente esiste sicuramente +un vettore $\Vec{v}$ tale che $\varphi_{f,\Vec{v}} = \varphi_f$. Allora, essendo $\varphi_f = \pm p_f$, deve valere +che $p_f = \pm \varphi_{f, \Vec{v}}$, ossia che la minima +combinazione lineare linearmente dipendente di $\Vec{v}$, ..., $f^k(\Vec{v})$ si può ottenere coinvolgendo almeno $n+1$ +termini (i.e.~con $k\geq n$). Allora i vettori +$\Vec{v}$, ..., +$f^{n-1}(\Vec{v})$ sono linearmente indipendenti, ed essendo +in totale $n$ formano una base di $V$. Pertanto $V = \Span(\Vec{v}, f(\Vec{v}), ...)$. +\end{proof} + +\begin{example} + Riprendendo l'esempio di prima, $\varphi_A(x) = (x-2)^2(x+4)$. Poiché $\deg p_A = 3$, allora $\varphi_A(x) - p_A(x)$. Allora per il teorema appena dimostrato deve + necessariamente esistere un vettore ciclico di $\RR^3$ su + $A$. \\ + + In effetti, posto $\Vec{v} = \begin{pmatrix} + 4 \\ -3 \\ 5 + \end{pmatrix}$, si ottiene che $\Vec{v}$, $A\Vec{v}$ e + $A^2\Vec{v}$ sono linearmente indipendenti, e sono + dunque una base $\basis$ di $\RR^3$. In particolare, la + matrice associata su questa base è la seguente: + + \[ M_\basis(A) = \begin{pmatrix} + 0 & 0 & -16 \\ + 1 & 0 & 12 \\ + 0 & 1 & 0 \\ + \end{pmatrix}, \] + + \vskip 0.1in + + proprio come ci aspettavamo che venisse da una delle osservazioni + iniziali, dal momento che $\varphi_A(x) = (x-2)^2(x+4) = x^3 - 12x + 16$. +\end{example} + + +\end{document} diff --git a/Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/personal_commands.sty b/Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/personal_commands.sty new file mode 100644 index 0000000..57dad17 --- /dev/null +++ b/Geometria I/LaTeX/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/personal_commands.sty @@ -0,0 +1,189 @@ +\ProvidesPackage{personal_commands} + +\usepackage{amsmath,amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsopn} +\usepackage{mathtools} +\usepackage{marvosym} +\usepackage{floatflt} +\usepackage{graphicx} +\usepackage{float} + +\hfuzz=\maxdimen +\tolerance=10000 +\hbadness=10000 + +\newcommand{\cororef}[1]{\textit{Corollario \ref{#1}}} +\newcommand{\exref}[1]{\textit{Esercizio \ref{#1}}} +\newcommand{\propref}[1]{\textit{Proposizione \ref{#1}}} +\newcommand{\lemmaref}[1]{\textit{Lemma \ref{#1}}} +\newcommand{\thref}[1]{\textit{Teorema \ref{#1}}} +\newcommand{\li}[0]{$\blacktriangleright\;\;$} + +\newcommand{\tendsto}[1]{\xrightarrow[\text{$x \to #1$}]{}} +\newcommand{\tendston}[0]{\xrightarrow[\text{$n \to \infty$}]{}} + +\setlength\parindent{0pt} + +% Personal commands +\newcommand{\card}[1]{\left|#1\right|} +\newcommand{\nsqrt}[2]{\!\sqrt[#1]{#2}\,} +\newcommand{\zeroset}{\{0\}} +\newcommand{\setminuszero}{\setminus \{0\}} + +\newenvironment{solution} + {\begin{proof}[Soluzione]} + {\end{proof}} + +\theoremstyle{definition} + +\let\abstract\undefined + +\newtheorem*{abstract}{Abstract} +\newtheorem{corollary}{Corollario} +\newtheorem*{definition}{Definizione} +\newtheorem*{example}{Esempio} +\newtheorem{exercise}{Esercizio} +\newtheorem*{goal}{Obiettivo} +\newtheorem{lemma}{Lemma} +\newtheorem*{remark}{Osservazione} +\newtheorem*{proposition}{Proposizione} +\newtheorem{theorem}{Teorema} + +\newcommand{\BB}{\mathcal{B}} +\newcommand{\HH}{\mathbb{H}} +\newcommand{\KK}{\mathbb{K}} +\newcommand{\ZZp}{\mathbb{Z}_p} + +\newcommand{\CCx}{\mathbb{C}[x]} +\newcommand{\FFpp}{\mathbb{F}_p} +\newcommand{\FFpd}{\mathbb{F}_{p^d}} +\newcommand{\FFpm}{\mathbb{F}_{p^m}} +\newcommand{\FFpn}{\mathbb{F}_{p^n}} +\newcommand{\FFp}[1]{\mathbb{F}_{p^{#1}}} +\newcommand{\KKx}{\mathbb{K}[x]} +\newcommand{\QQx}{\mathbb{Q}[x]} +\newcommand{\RRx}{\mathbb{R}[x]} +\newcommand{\ZZi}{\mathbb{Z}[i]} +\newcommand{\ZZom}{\mathbb{Z}[\omega]} +\newcommand{\ZZpx}{\mathbb{Z}_p[x]} +\newcommand{\ZZsqrt}[1]{\mathbb{Z}[\sqrt{#1}]} +\newcommand{\ZZx}{\mathbb{Z}[x]} + +\newcommand{\ii}{\mathbf{i}} +\newcommand{\jj}{\mathbf{j}} +\newcommand{\kk}{\mathbf{k}} + +\newcommand{\valalpha}{\varphi_\alpha} +\newcommand{\Frob}{\mathcal{F}} +\newcommand{\Frobexp}{\mathcal{F}{\mkern 1.5mu}} + +\newcommand{\dual}[1]{#1^{*}} +\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} +\newcommand{\M}[1]{\mathcal{M}_{#1}\left(\KK\right)} +\newcommand{\nsg}{\mathrel{\unlhd}} +\renewcommand{\vec}[1]{\underline{#1}} + +\newcommand{\hatpi}{\hat{\pi}} +\newcommand{\hatpip}{\hat{\pi}_p} + +% evan.sty original commands +\newcommand{\cbrt}[1]{\sqrt[3]{#1}} +\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} +\newcommand{\ceiling}[1]{\left\lceil #1 \right\rceil} +\newcommand{\mailto}[1]{\href{mailto:#1}{\texttt{#1}}} +\newcommand{\eps}{\varepsilon} +\newcommand{\vocab}[1]{\textbf{\color{blue}\sffamily #1}} +\providecommand{\alert}{\vocab} +\newcommand{\catname}{\mathsf} +\providecommand{\arc}[1]{\wideparen{#1}} + +% From H113 "Introduction to Abstract Algebra" at UC Berkeley +\newcommand{\CC}{\mathbb C} +\newcommand{\FF}{\mathbb F} +\newcommand{\NN}{\mathbb N} +\newcommand{\QQ}{\mathbb Q} +\newcommand{\RR}{\mathbb R} +\newcommand{\ZZ}{\mathbb Z} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\val}{val} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} + + +%From Kiran Kedlaya's "Geometry Unbound" +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +\newcommand{\norm}[1]{\left\lVert #1 \right\rVert} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} + +% From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} +\newcommand{\defiff}{\overset{\mathrm{def}}{\iff}} + +% From the USAMO .tex files +\newcommand{\dg}{^\circ} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\DeclareMathOperator{\Char}{char} +\DeclareMathOperator{\Dom}{Dom} +\DeclareMathOperator{\Fix}{\textit{Fix}\,} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\existsone}{\exists !} +\DeclareMathOperator{\Span}{Span} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Imm}{Imm} +\DeclareMathOperator{\Ker}{Ker} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\MCD}{MCD} +\DeclareMathOperator{\Mor}{Mor} +\DeclareMathOperator{\mcm}{mcm} +\DeclareMathOperator{\Sym}{Sym} +\newcommand{\basis}{\mathcal{B}} +\DeclareMathOperator{\tr}{tr} + +\let\oldemptyset\emptyset +\let\emptyset\varnothing + +\let\oldcirc\circ +\let\circ\undefined +\DeclareMathOperator{\circ}{\oldcirc} + +\let\oldexists\exists +\let\exists\undefined +\DeclareMathOperator{\exists}{\oldexists} + +\let\oldforall\forall +\let\forall\undefined +\DeclareMathOperator{\forall}{\oldforall} + +\let\oldnexists\nexists +\let\nexists\undefined +\DeclareMathOperator{\nexists}{\oldnexists} + +\let\oldland\land +\let\land\undefined +\DeclareMathOperator{\land}{\oldland} + +\let\oldlnot\lnot +\let\lnot\undefined +\DeclareMathOperator{\lnot}{\oldlnot} + +\let\oldlor\lor +\let\lor\undefined +\DeclareMathOperator{\lor}{\oldlor} \ No newline at end of file diff --git a/Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo.pdf b/Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. 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