From b91e2c98b00a868085a9f71066de1950ac4ab7be Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 30 Jun 2023 09:54:23 +0200 Subject: [PATCH] feat(geometria/schede): le matrici elementari generano GL(n, K) --- Geometria 1/Scheda riassuntiva/main.pdf | 1858 +++++++++++------------ Geometria 1/Scheda riassuntiva/main.tex | 36 +- 2 files changed, 933 insertions(+), 961 deletions(-) diff --git a/Geometria 1/Scheda riassuntiva/main.pdf b/Geometria 1/Scheda riassuntiva/main.pdf index b679a25..26f5d55 100644 --- a/Geometria 1/Scheda riassuntiva/main.pdf +++ b/Geometria 1/Scheda riassuntiva/main.pdf @@ -415,41 +415,43 @@ endobj endobj 180 0 obj << -/Length 7446 +/Length 7612 /Filter /FlateDecode >> stream -x]ݓ㶑߿BdM C|vkSNU<wL"ǢK|5@Hj4YP nǯ~>۟|wo\Q*\}n)7շ?}nw17Buvsϫ|yYva_l}0:[(U1ѣǫ;#FH;<]o*O֛i'͸z&I‹v ?T(2|J+.12h3G={52"i꽆 Hjuޒ)Tm~QɀD737T?nķ+;JhDK#.Ep /|t,~:+Jo!Ç 3M!U]e -P|_yts]woLniY?llZ>IMNj vbg'*tvC !{j7: # v}׈b!)E$y7DcϢfu?x(a* oyĨT#PJyzNh]LxU4/K7_yЫ#5#S*#XR88"B.b[Gywq@Rg7ЀV -_r*: j)[Y98<gNf{GVԊ]!{cdNN瑆 fzUD,N$%!0H`F(:swoKD( -Xo|GI;+Y<]C:5nUgE-K𣩼ai US\ű< -11ǜ -X([ƸNJKp~QQ5A{l&ai`a;Pg߼1.Y)`S1j{6̃;ׯ&#8]wPӪN2ؚXY\YF -o]Z9FI%0[}tLNJMdVE^*ՙҐ| -ۥWI@tW~P's#ړڃkt - VOQʦB\hLdI?Wrܿ@殣8pr68Q08NƝ"k^§&DpW`lM~SlodҭJ͐i@Rl*uigGl4T̆. .]i/l#D$cP}zAtwr/`önX/Y0\ta];/ , !dVB&xG2(ބ}Qs,̛S s2CP엢r?&]d]xHRF ֙<9ĴĴ=\J>ĺKjEv_W]+KR@ºZ;?uBOPT̂NjܽpeٹۙOǼ,Q=쳙s5H:6(ÑW{"f~>Tdﰜ[RQrXЋvdl1mu% .ԡ~+{;1a:r|6Zw >Z~PTM﬚A 9a39ib}8 T܎P:Gé1Uq)`oO<-<"%p&Ĕ^KM `^5a&&i.0V(`,FTXk+UcWcWcBS,Yq 0JW"ud{LPQS\Xyxr EwJ;X >4-dlk8Uxqo΃mrc %#N z 4Jy`O-'P.MuaH<$L|"H:/LYG&h1XG17L뇬{ -}K"U9R9<Q'?4q)7F̣7ngMp]`P=7Bx|6cCÞ/q0}4?cvNҐᾹx;1N4pmic;L4JNmx L/Ma v$><^{p8->-.چ 1te0`΁(L:/<:J$D渖8sPn1y#)M45IL}X9-;) -* Tzxs(tFr)#SXeoP."N!$2#>~{p€xڛxhbnKd=õc=8Eh$2E,ˢʳs~]| jV(&!7E8LUgn(wsCq_ebBwug -@ʘ0wz[rӠhʺE”Nؐ7Ɔ'gM~JT әI~Kvk@ug9;x&ч(SbGZSsN \߽U 1P۬C\/$,&~ WV2c- l;ļ -9:/ʸ+R8$J/`7`77G?-4 ;-G3y^ --rHEtIV[4z_bU~5XLT[Ā:#NtQׁ"ufsرamDžU+j=\x9VE%,o`wK?!k7aS+ֆ[$MzvD{7tx -K1 7UpђPᎆCŝ1e /@BGjB=QvEy7?h0,0ke^mr۴a&<ч}Qmm9y=G -h󕹞\yڛva {xT.dA  4DPD.إBnL#rVNNMtg8^rf -(inh6s7SBX MBRF] syXRfX#Ү8GEJ+/ꇂ'Jê3+QωIu6α$ \+ez|Ŝ<CcǰR=>mG"WVԓa?NX@2 *$qw;zR-vYLN~,zLt$&5&f͆cU}ai#u䊝wPɯ&AvXDIE`<5OLs MH2KVů!}oyGWL0{Y}6u7ƛ^a݌vif҄_w@9e]U cpNٶ6a[G16nЫ;4B$j9|TJItœ$љ''Tt{~pA\M "ո[jJmPT(h5>H%$-ewB"5&idmVMxp"=Zjvu!`PB/ -,)':IĆi !qq T41~1j28Xzp" -C&wv_a{u?A>ł~ebg#yJ$Z,C57uz Y?,>n_Ƨ%S֐Jx4HKAÜ8/Yd2Aߐ.;KoŬt[Z2䩤mC`fYtgU~|%Ap0AvivYUs/,,`0K9*Z]köL"gS`,g͡8Ǜœqشu6O|uiL6ZzOe-΂D!Lb m{ٺ͍`?>G:6n)|;$$4C4R h8:ᩎJ%AM#5Bx'2`;"h1btȼS\e_tUgii1aHp J`!AջpbwTgkՇcOYj6FkuC$eDWB<=ЉYӿ/M% -Jᦳ]D\/+WFD:l g[P[,)xhI2 5S9\ b7P5VD`s*ےIg!&,u)jl(lEepgPN{Yo] ^>SA_k"~ff&RnLn$-S/>ׯkBPVYdJ.άΝsSNgD!"cny[4i9ۆ>r@U qX?˟u5d#/q`G}}SxMQVH7Bݬ3*ꩾ׆렸F/pD$NV`d ]6$ -~&N=p>ސN7^&4@h,d33MnMM7Qy\#DBUc6Qugiۄ -cO |FSjKS*: %Fo*T׬;?P6. 9W{bwu:b)I0oۯ~]8Fn}[BRzv=oaKЀMp+0#eo& pt.n% pC?^/ :{Y|#g-gk e%Whi=ms Z4fƒςǑsE#Bw? -RpK/ݭDY{q^k6} F=mKԺm*W6T䚦HKr"g{,/Wz؝CDFUD(%wu FN?0arRT"%})eŧ}L hKk)C0ǁ5e`o0*z}eף5/#}RWqA̦dc W! 2z$Po> LT8Lu9J+8{39,+|v?7 ъb <)]h~<:[ 4^ 6 tKAlnv3g!Ō%#/ɺ0ݹ{m= _ ^([DZx /gZ&;X^W!dY/z(2]x&j]mc2%sDvECyjSv#(Iwo3k7&`tAa0˓Afγ;VتRϼsN'F{aWq<7\NDA̬4!e9h0m۰f >]0#괪 4Bq[ NBoy~P}s.r7\*un3e-]%Vy=G?[kQ@ 7ĕe0ՌC0 ZB) Q -*1iTEd+߸>t̅f f~h4vB۫;NBn/GLK! z9W˄)I/՝;&,tY** -ӗ[_( s܆-\܆x8oUDUlFDWs}CRB)RU+*۾z`e1ҁ= u Q$T|u1sڇ)AqXc} ŵkWX6?>iP}Nפe8ud^e])ԛMfqGq*5tڇ><Cy@>fM_d2!}O'ʛ/&lmg̲ot# eB;5AGw kZK&%Uu긋Kb2kϛ(#fuo - ;%>ճHq**Fw/B=[v䊽[s| 8%X{ {aȊOI;HQ,Z'j` n|/gmso3{`Ĉ*޶i郭Ksq[@;suiB_=>DQv<{i:r3>z tO On!3I D&4J@ɯpj=Ո¶1) HөHP*rujV2qmG m AC>/7| 32 -z\?E ?W=)YD~~*Y{kƯM֪0M ѝ{83%o1v 0x+Z]K(C%iL)[Ӡ7rvTh1Eyh|j݌2}c ]$|O2 +x]Y䶑~_pMܤڈH#GGlXӢ"[uhm,խyf@"痉&El~ů_3yFEHLݼ|Ӿ~6Y}Sz+Zek}^,o|2oˎbSoatvTD߬ !owyU64]}o9UYw=-YdOqM(Ͱ/"tPd&? sW$=],bdєgϚ'8 G{wkdD{ ?!ɡ2[R?oNz$꽙qby#_YSB#uq)kxSӭHWYWz#>>4Oi + +>P. 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+0000431215 00000 n +0000431411 00000 n +0000431605 00000 n +0000431804 00000 n +0000432004 00000 n +0000432197 00000 n +0000432305 00000 n +0000432416 00000 n +0000432529 00000 n +0000432645 00000 n +0000432735 00000 n +0000432773 00000 n +0000432901 00000 n trailer << /Size 408 /Root 406 0 R /Info 407 0 R -/ID [<25D4570036AACA5FCCA7974762791155> <25D4570036AACA5FCCA7974762791155>] >> +/ID [ ] >> startxref -431429 +433175 %%EOF diff --git a/Geometria 1/Scheda riassuntiva/main.tex b/Geometria 1/Scheda riassuntiva/main.tex index 04ff951..6fcbf68 100644 --- a/Geometria 1/Scheda riassuntiva/main.tex +++ b/Geometria 1/Scheda riassuntiva/main.tex @@ -562,10 +562,11 @@ \end{enumerate} A queste operazioni è associato il prodotto a sinistra per delle particolari matrici. - In particolare, l'operazione di scambio della riga $i$-esima con quella $j$-esima corrisponde alla moltiplicazione a sinistra per la matrice $S_{i,j}$, dove: + In particolare, l'operazione di scambio della riga $i$-esima con quella $j$-esima corrisponde alla moltiplicazione a sinistra per la matrice $S_{i,j}$, detta matrice + elementare di permutazione, dove: \[ S_{i,j}=I_n-E_{i,i}-E_{j,j}+E_{i,j}+E_{j,i}. \] - Per esempio, scambiare due righe in una matrice $2 \times 2$ corrisponde a + In particolare, la stessa matrice $S_{i,j}$ si ottiene scambiando la riga $i$-esima e la $j$-esima. Per esempio, scambiare due righe in una matrice $2 \times 2$ corrisponde a moltiplicare a sinistra per $S_{1,2}$, dove: \[S_{1,2}=\begin{pmatrix} @@ -573,19 +574,26 @@ 1 & 0 \\ \end{pmatrix}\] - All'operazione di moltiplicazione di una riga per uno scalare non nullo corrisponde invece la matrice: + All'operazione di moltiplicazione della riga $i$-esima per uno scalare $\lambda \neq 0$ corrisponde invece la matrice $M_{i, \lambda}$, detta elementare di dilatazione, dove: \[ M_{i,\lambda} = \Matrix{I_{i-1} & \rvline & 0 & \rvline & 0 \\ \hline 0 & \rvline & \lambda & \rvline & 0 \\ \hline 0 & \rvline & 0 & \rvline & I_{n-i} }. \] - All'operazione di somma di un multiplo non nullo di una riga ad un'altra riga distinta corrisponde invece la matrice: + All'operazione di somma della riga $j$-esima moltiplicata per $\lambda \neq 0$ alla riga $i$-esima corrisponde invece la matrice $M_{i,j,\lambda}$, detta elementare di trasvezione (o di tosatura, dall'inglese \textit{shear matrix}), dove: \[M_{i,j,\lambda}=I_n+\lambda E_{i,j}.\] - - Tutte queste matrici sono invertibili ed in particolare valgono le seguente relazioni: + Se $\lambda \neq 0$, tutte queste matrici sono invertibili ed in particolare valgono le seguente relazioni: \begin{itemize} - \item $S_{i,j}^{-1}=S_{i,j}$, - \item $M_{i,\lambda}^{-1}=M_{i,\frac{1}{\lambda}}$, + \item $S_{i,j}^{-1}=S_{i,j}$, da cui si osserva che l'inversa di una matrice elementare di permutazione è ancora una matrice dello stesso tipo), + \item $M_{i,\lambda}^{-1}=M_{i,\frac{1}{\lambda}}$, come sopra, + \item $M_{i,j,\lambda}^{-1}=M_{i,j,-\lambda}$, come sopra, \item $M_{i,\lambda} = M_{i, i, \lambda-1}$, - \item $M_{i,j,\lambda}^{-1}=M_{i,j,-\lambda}$. + \item le matrici elementari generano il gruppo delle matrici invertibili $\GL(n, \KK)$, ossia ogni matrice + invertibile si scrive come prodotto di matrici elementari (è sufficiente + applicare l'algoritmo di eliminazione di Gauss per righe su $A \in \GL(n, \KK)$ e + osservare che ne deve risultare obbligatoriamente una matrice diagonale che, + normalizzata sugli elementi, restituisce esattamente $I_n$; allora poiché applicare l'algoritmo + equivale a moltiplicare a sinistra per delle matrici elementari $\mathcal{E}_1$, ..., $\mathcal{E}_k$, si verifica che $\mathcal{E}_1 \cdots \mathcal{E}_k A = I_n \implies A = \mathcal{E}_k \inv \cdots \mathcal{E}_1 \inv$, dove si + conclude la dimostrazione osservando che l'inversa di una matrice elementare + è ancora una matrice elementare). \end{itemize} Queste operazioni non variano né $\Ker A$ né $\rg (A)$. Permettendo di variare $\Ker A$ si possono effettuare le stesse medesime operazioni @@ -593,8 +601,7 @@ invariato $\Im A$, e quindi $\rg (A)$): tali operazioni corrispondono a moltiplicare a destra per una matrice invertibile, analogamente a come accade per le righe. \\ Le matrici per cui si moltiplica a destra per operare sulle colonne sono esattamente le stesse matrici impiegate - per le operazioni di riga, sebbene trasposte. In particolare le matrici di scambio - di riga e di moltiplicazione per uno scalare coincidono. Pertanto, se $A$ è una matrice simmetrica (i.e.~se $A \in \Sym(n, \KK)$), operare mediante le stesse + per le operazioni di riga, sebbene trasposte (e quindi sono ancora matrici elementari). In particolare le matrici elementari di permutazione (per scambiare le righe) e di dilatazione (per moltiplicare una riga per uno scalare non nullo) coincidono. Pertanto, se $A$ è una matrice simmetrica (i.e.~se $A \in \Sym(n, \KK)$), operare mediante le stesse operazioni sulle righe e sulle colonne permette di individuare matrici congruenti ad $A$. @@ -2121,7 +2128,7 @@ Si definisce \textit{stabilizzatore} di $x \in X$ il sottogruppo di $G$ $\Stab_G(x)$ tale che: - \[ \Stab_G(x)= \{g\in G | g \cdot x=x\}, \] + \[ \Stab_G(x)= \{g\in G \mid g \cdot x=x\}, \] dove si scrive $\Stab(x)$ per indicare $\Stab_G(x)$ qualora non fosse ambigua l'azione a cui ci si riferisce. \\ @@ -2143,7 +2150,7 @@ \begin{enumerate} \item $\GL(n,\KK)$ opera su $M(n,\KK)$ tramite la similitudine e le orbite sono le classi di matrici simili, rappresentate dalle forme canoniche di Jordan, \item $\GL(n,\KK)$ opera su $\Sym(n,\KK)$ tramite la congruenza e le orbite sono le classi di matrici congruenti, rappresentate in $\RR$ dalle matrici diagonali con $1$, $-1$ e $0$ come elementi, e in $\CC$ dalle stesse matrici rappresentanti delle classi - di equivalenza della SD-equivalenza, + di equivalenza della SD-equivalenza, ossia le matrici del tipo $I_r^{m \times n}$, \item $O_n$ agisce naturalmente su $\RR^n$ e l'orbita di $\vec x \in \RR^n$ è la sfera di raggio $\norm{\vec x}$ secondo il prodotto scalare standard di $\RR^n$. \end{enumerate} @@ -2167,7 +2174,8 @@ Si dice che $G$ opera in maniera \textit{semplicemente transitiva} su $X$ se opera transitivamente e liberamente su $X$; in tal caso si dice che $X$ è un insieme $G$-omogeneo principale. Equivalentemente $G$ opera in maniera semplicemente transitiva se $\exists x\in X$ t.c.~$g\rightarrow g \cdot x$ è una bigezione. - Se $X$ è un insieme $G$-omogeneo e $G$ è abeliano, allora $G$ agisce fedelmente su $X$ $\iff$ $X$ è $G$-insieme omogeneo principale. + Se $X$ è un insieme $G$-omogeneo e $G$ è abeliano, allora $G$ agisce fedelmente su $X$ $\iff$ $X$ è $G$-insieme omogeneo principale (per dimostrare l'implicazione a destra è sufficiente mostrare che, se $x \in X$, $g \in \Stab(x) \implies f_g = \IdV{X}$, da cui si conclude + che $g = e$ per la fedeltà dell'azione). \subsection{Proprietà generali di uno spazio affine} Si dice spazio affine $E$ un qualcune insieme $V$-omogeneo principale, dove