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hermitiano/Appunti originali/2023-03-27, 31, Proprietà e teoremi principali sul prodotto scalare/main.tex @@ -86,9 +86,9 @@ \end{theorem} \begin{proof} - Si dimostra il teorema per induzione su $n := \dim V$. Per $n \leq 1$, la tesi è triviale (ogni base è - già una base ortogonale). Sia - allora il teorema vero per $i \leq n$. Se $V$ ammette un vettore non isotropo $\vec w$, sia $W = \Span(\vec w)$, e si consideri la decomposizione $V = W \oplus W^\perp$. Poiché $W^\perp$ ha dimensione $n-1$, per ipotesi induttiva + Si dimostra il teorema per induzione su $n := \dim V$. Per $n \leq 1$, la tesi è triviale (se esiste una base, tale base è + già ortogonale). Sia + allora il teorema vero per $i \leq n$. Se $V$ ammette un vettore non isotropo $\vec w$, sia $W = \Span(\vec w)$ e si consideri la decomposizione $V = W \oplus W^\perp$. Poiché $W^\perp$ ha dimensione $n-1$, per ipotesi induttiva ammette una base ortogonale. Inoltre, tale base è anche ortogonale a $W$, e quindi l'aggiunta di $\vec w$ a questa base ne fa una base ortogonale di $V$. Se invece $V$ non ammette vettori non isotropi, ogni forma quadratica è nulla, e quindi il prodotto scalare è nullo per la proposizione precedente. Allora in questo caso @@ -121,27 +121,30 @@ in essi sia diversa da zero. \end{proof} - \begin{remark} - Si possono effettuare alcune considerazioni sul teorema di Sylvester - complesso. \\ - + \begin{remark}\nl \li Si può immediatamente concludere che il rango è un invariante - completo per la congruenza in un campo in cui tutti gli elementi + completo per la congruenza in un campo $\KK$ in cui tutti gli elementi sono quadrati, ossia che $A \cong B \iff \rg(A) = \rg(B)$, se $A$ e - $B$ sono matrici simmetriche: infatti - ogni matrice simmetrica rappresenta una prodotto scalare, ed è + $B$ sono matrici simmetriche con elementi in $\KK$. \\ + + Ogni matrice simmetrica rappresenta infatti un prodotto scalare, ed è pertanto congruente ad una matrice della forma desiderata nell'enunciato del teorema di Sylvester complesso. Poiché il rango è un invariante della congruenza, si ricava che $r$ nella forma della matrice di Sylvester, rappresentando il rango, è anche - il rango di ogni sua matrice congruente. In particolare, se due + il rango di ogni sua matrice congruente. \\ + + In particolare, se due matrici simmetriche hanno lo stesso rango, allora sono congruenti alla stessa matrice di Sylvester, e quindi, essendo la congruenza una relazione di equivalenza, sono congruenti a loro volta tra di loro. \\ - \li Due matrici simmetriche con stesso rango, allora, non solo + + \li Due matrici simmetriche in $\KK$ con stesso rango, allora, non solo sono SD-equivalenti, ma sono anche congruenti. \\ + \li Ogni base ortogonale deve quindi avere lo stesso numero - di elementi nulli. + di vettori isotropi, dal momento che tale numero rappresenta + la dimensione del radicale $V^\perp$. \end{remark} \begin{definition}[somma diretta ortogonale] @@ -231,19 +234,19 @@ \end{enumerate} \end{solution} - \begin{definition} + \begin{definition} [indici e segnatura] Data una base ortogonale $\basis$ di $V$ rispetto al prodotto scalare $\varphi$, si definiscono i seguenti indici: - \begin{align*} \iota_+(\varphi) &= \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} > 0 \}, &\text{(}\textbf{indice di positività}\text{)} \\ \iota_-(\varphi) &= \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} < 0 \}, &\text{(}\textbf{indice di negatività}\text{)}\\ \iota_0(\varphi) &= \dim V^\perp. &\text{(}\textbf{indice di nullità}\text{)} \end{align*} - Quando il prodotto scalare $\varphi$ è noto dal contesto, si omette - e si scrive solo $\iota_+$, $\iota_-$ e $\iota_0$. In particolare, + Quando il prodotto scalare $\varphi$ è noto dal contesto, si + semplifica la notazione + scrivendo solo $\iota_+$, $\iota_-$ e $\iota_0$. In particolare, la terna $\sigma(\varphi) = \sigma = (i_+, i_-, i_0)$ è detta \textbf{segnatura} del prodotto $\varphi$. \end{definition} @@ -272,7 +275,8 @@ la cui matrice associata del prodotto scalare è come desiderata nella tesi. \\ - Sia ora $a$ il numero di vettori della base con forma quadratica + Sia ora $\basis$ una qualsiasi base ortogonale di $V$. + Siano inoltre $a$ il numero di vettori della base con forma quadratica positiva, $b$ il numero di vettori con forma negativa e $c$ quello dei vettori con forma nulla. Si consideri $W_+ = \Span(\vv 1, ..., \vv a)$, $W_- = \Span(\vv{a+1}, ..., \vv b)$, $W_0 = \Span(\vv{b+1}, ..., \vv c)$. \\ @@ -294,34 +298,42 @@ \begin{definition} Si dice \textbf{base di Sylvester} una base di $V$ tale per cui la matrice associata di $\varphi$ sia esattamente nella forma - vista nella dimostrazione del teorema di Sylvester. Analogamente + vista nell'enunciato del teorema di Sylvester. Analogamente si definisce tale matrice come \textbf{matrice di Sylvester}. \end{definition} \begin{remark} \nl - \li Si può dunque definire la segnatura di una matrice simmetrica - come la segnatura di una qualsiasi sua base ortogonale, dal - momento che tale segnatura è invariante per cambiamento di base. \\ - \li La segnatura è un invariante completo per la congruenza nel caso reale. Se infatti due matrici hanno la stessa segnatura, sono - entrambe congruenti alla matrice come vista nella dimostrazione - della forma reale del teorema di Sylvester, e quindi, essendo + \li Come conseguenza del teorema di Sylvester reale, si osserva che la segnatura di una matrice simmetrica reale + è invariante per cambiamento di base, se la base è ortogonale. \\ + + \li La segnatura è un invariante completo per la congruenza nel caso reale. Se infatti due matrici hanno la stessa segnatura, queste sono + entrambe congruenti alla stessa matrice di Sylvester, e quindi, essendo la congruenza una relazione di equivalenza, sono congruenti tra loro. Analogamente vale il viceversa, dal momento che ogni - base ortogonale di due matrici congruenti devono contenere gli + base ortogonale di due matrici congruenti deve contenere gli stessi numeri $\iota_+$, $\iota_-$ e $\iota_0$ di vettori di base con forma quadratica positiva, negativa e nulla. \\ + \li Se $\ww 1$, ..., $\ww k$ sono tutti i vettori di una base - ortogonale $\basis$ con forma quadratica nulla, si osserva che $W = \Span(\ww 1, ..., \ww k)$ altro non è che $V^\perp$ stesso. Infatti, come + ortogonale $\basis$ con forma quadratica nulla, si osserva che $W = \Span(\ww 1, ..., \ww k)$ altro non è che $V^\perp$ stesso. \\ + + Infatti, come visto anche nella dimostrazione del teorema di Sylvester reale, vale - che $\dim W = \dim \Ker (M_\basis(\varphi)) = \dim V^\perp$. Inoltre, - se $\w \in W$ e $\v \in V$, $\varphi(\w, \v) = \varphi(\sum_{i=1}^k + che $\dim W = \dim \Ker (M_\basis(\varphi)) = \dim V^\perp$. + Sia allora la base $\basis = \{\ww 1, \ldots, \ww k, \vv{k+1}, \ldots, \vv n\}$ un'estensione di $\{\ww 1, \ldots, \ww k\}$. Se $\w \in W$ e $\v \in V$, $\varphi(\w, \v) = \varphi(\sum_{i=1}^k \alpha_i \ww i, \sum_{i=1}^k \beta_i \ww i + \sum_{i=k+1}^n \beta_i \vv i) - = \sum_{i=1}^k \alpha_i \beta_i q(\ww i) = 0$, e quindi - $W \subseteq V^\perp$, da cui si conclude che $W = V^\perp$. \\ - \li Vale in particolare che $\rg(\varphi) = \iota_+ + \iota_-$, mentre - $\dim \Ker(\varphi) = \iota_0$, e quindi $n = \iota_+ + \iota_- + \iota_0$. \\ - \li Se $V = U \oplusperp W$, allora $\iota_+(\varphi) = \iota_+(\restr{\varphi}{V}) + \iota_+(\restr{\varphi}{W})$, e - analogamente per gli altri indici. + = \sum_{i=1}^k \alpha_i \beta_i q(\ww i) = 0$ (dove $\alpha_i$ e $\beta_i \in \KK$ rappresentano la $i$-esima coordinata di $\w$ e $\v$ nella base $\basis$), e quindi + $W \subseteq V^\perp$. Si conclude allora, tramite l'uguaglianza + dimensionale, che $W = V^\perp$. \\ + + \li Poiché $\dim \Ker(\varphi) = \iota_0$, vale in particolare che $\rg(\varphi) = n - \iota_0 = \iota_+ + \iota_-$ (infatti vale che $n = \iota_+ + \iota_- + \iota_0$, dal momento che $n$ rappresenta il numero di elementi di una base ortogonale). \\ + + \li Se $V = U \oplusperp W$, allora $\iota_+(\varphi) = \iota_+(\restr{\varphi}{V}) + \iota_+(\restr{\varphi}{W})$. + Analogamente vale la stessa cosa per gli altri indici. Infatti, + prese due basi ortogonali $\basis_U$, $\basis_W$ di $U$ e $W$, + la loro unione $\basis$ è una base ortogonale di $V$. Pertanto + il numero di vettori della base $\basis$ con forma quadratica positiva + è esattamente $\iota_+(\restr{\varphi}{V}) + \iota_+(\restr{\varphi}{W})$. \end{remark} \begin{definition}[isometria tra due spazi vettoriali] diff --git a/Geometria 1/Spazi affini/2023-04-28, Indipendenza e applicazioni affini/main.pdf b/Geometria 1/Spazi affini/2023-04-28, Indipendenza e applicazioni affini/main.pdf index fa1966214d4e34fcacf3c3b34648cf1fc5a56d21..96fde56ab2a6ed9822baa8190440326675a036b6 100644 GIT binary patch delta 5753 zcmah~c{r4R_qOk42@x{Zh-%hnozV zQOx~v@o9v9rtDM8N>R6kTD|$0tNoG{liINhLGo^gbXEoCp64$=)yLCBAs9)2bK8>_ z%*c{F&cMd^YxM0=XJw&|1CK(Hj>L%wU4cwHCyL~oUYnO;wr?7^>~ln|Lam(9Hfr^D zC&Q*MN{4d$MNu{)4<8u0G(cbWv4{OfqJ&)tF8=_n?fbdGL|5PQ`J>gbU1`%5?q{4` z%2Ez`4!SNpOPXM&6%BKp+!4@-8JpYEY+0R((;Wkn@+T(Ly>o^j`PR)rK{HvPHN!{G z!TaQNN#~M-skn~XsfS}Obx=vth^oeaW|G9UizIR23 zzT_~nBKS~QdY8qh9n@@nFQYc}>cWBVk%NgJ>E@^D-&FeDk^eILGNkwF1+<cl_hg9{i%k_A*=9{e1xU0bO!Nt$Q9Wx^bh2^JK?sp3gU)@UV7m_uMm+5V>*K zLia<}Msw~{6uS7Lu^}D()vP0}`7sS@T$$gWeGf_g?!o9%sNJ*dqqmp$Bcknft?V-f zAPnJ1(wAV}DU6CRgM9C5M5w8La*!0;KwGtL&(V~dt9bf~kZuRgzfn`X*1%gB^d&~m z7|%5jKHPlaVb;n>%R&}Q-=bGzMos~5M3NRcdlUUZX(RaI=mcQae+>C}5m6H_iYno) zIK_;Ke&D=;8CrhL+KgjmyDcx#WAS*1Mzv&2HqDysp#6x<4K3e&;P`2z(tQ8epd031 ze2J6o$7%O2ozqgYZ!LYpZTz*P&(=4%9QPvohIIEe77X1=x}h)NPigR5MxTfu>&p;} zvsyQp52)99bS-6dyl>AEgD3~Oh6VVTi*d z8&+a5!kGbY-LCQ`z^H{SMhh+Hd(;l${j( z_h)XrWZM~3g>(*xy-?u~WG=ooW6@YK=MD`uzH4-QezWw%qQT*~W_4~-tKSP{pY4hk z(ri3BN$*>uxe8p}&Cw&jb@I8S0Hknl?Xd@&s8f@c2T$sG;AJ0e zL->j1csgy5GTETxc1kL$QI}eT=8wHuTr{4{AaZjtSSMOo_951^m79pY*Cbw7H4+^R z?H9f55EYclh5hPx^GFfXI$sTvb)vA=GtYg~uX2f_Tt~LXUWS=W7ovNt z&qHZqtFVjHK88z|O~VF{2A#0e6S%?Ooz8Xh0^l>=bl&E2M9!G5-le7U5zKmJbf?np z?p+O0fN0hy%pM0NDVf>oa_B0&%NX?GPe~oKd(2x;j-I<`vysV!Gbtsl8$ZxD8ypswOACX=#f5 z{A;+}Qh$l-RwK~B-LuhhACzNSUT)Z{98@8m`qexe9dMD?E4On*S+B0q{CoQQ*dvuk z97`u5P5IW!ZDq#kb;JXR1CXk0mLlOfm!*X}?v`YbrWpqsBBIoX+qj#Z46EiM6-1bZ z<4zg(>5Z)AN!xT;8O)T*98ox{K*aa@rdOz>U*J83mVXbhiHFv{uQ*PY2%Ms6yj8W< z0p(-}Y4SQfvA20+|FzTJN&chVlhs#dRSTgmX2oSZ6r;<<`?8I6J9=NN<|LN~%YBIK z(dwT{aF}WL`(`hDqRiDdu|!2(p7xr@fGr^E7NyJTpC`ep(5~HBbmj2DC=X(vR6b>t z$OY7X+Gc!j{phkt^^+;Fa@Bk@5$pWVT7i&woXkGoSgyvvhoo`p8uk10W*6iStzgnUCunCs0TxWaQCOma-pI}Ip* zy-PZ1s3Dkknrlj>%WsgGM=?kZFQd}98`43jL%yKY3xp~=9<8eZ#{cs`o_*21r z`L9(y?4<<(ZeAgBPOQuaWQ`|+Z*o7&SbUXqH68JqRAzu^XJ3^${E@`XPAb3n7i2~& zWutLMEax8Bs%MlCs?T=TOUqBY6lyU^lV#GtRh(#Z&7~+@@%^UIwIU1I}#o(^#9zBLgl6B_Ai}5!_)^Z^xGvb25!e;q_mNRRl^r)4C>BWvw`r7qnW-P8u zc(F(W}1VY{dXgIo_|C ziUJ$+Tg;S!obW&*WjSMPGRrX(D2g{tRk9bmomE_ zQ%ccdZCrBe{ak(HKN6FEyvrT;84-+%e-`6h(-UmtvPYG{mZECp($zO+&~@UG^nuFu;v4MOptYFEV=j_{ zpBX9&k0+SRdNlw}PyF>d#aG02TRTX5Z1NA~$U*#8@QFhlIUl(+$`}&YocJg}^ zNj+P62Pf|x6*9}_nEO)4HQawBTA0)%qE>pLF)dcMt|Z+~CpL9@*0c$Wt){$o<-AlD zvZ&)(JF4h5@(T4?Fj32@9wN#O5lk=HdAm3i4S!FG-Eq3=N#E3bJUeL;23s+&fK$lm1oms?v=Z**jo+Fi#6nloum-#y6CeNv^^_O9nvR{!2ar|W0hD~k6iDM~+`_dQcH=v`<*$HV(ASzX3Q1r>w* zQQ2tl(ye2jSE+g*_K@==wB-`2$y#BkH}9;$r^6Nty=ME zwBCC6ft~8r>M!&Kymdd~Is}RlUz&4k0*)0kt|S*PDX*GubER4%J*;Z>3jxb(tq*EaCo+Jq^>T8Sk@Y@eKP< zC@b9xW^}3*_4J-mbB_#i{}DJ5z&gm@d$wRx^tR$>zi4iieL-W^x5un$OUA6}{?4Pv zqww(oob-r)|48SguZ3_3F{7&wN*VPOxV2zUU$D-MIjpy9b#EE4|*c8tmxBw*m| zKoG$E8Hl8IjRL@ud))LVY58CcAkjDsRoMw1`a=*b4T&ZYU^S6c3{Utaz;FI&9Ds&# zKw{7s986FuhQ`8pAu&iKi24t&{}L39qx$+&>)-AoF&HFVyq_2VZw7iQM=gT0B~1f9R7dA;qh?6F*p#liyV%C1)x8=p*8~ms9kd5 z(YurJAbebq7!V8WQUL^aD@efN|FIqVwMa-DngIL*{>Cs=PhbrKREyyaQil)t<>&AD zz*8msQ)(m(+8 zLF!Z^sQ&-gz@gC8djeyJ!T|pZ!@?P(uvjd7;HVe?w*-a55|F=bf_~jG91a7Q3k9G- z44esop$Z0Lg2!SAuz^G2u>kIOkzX}HG*uf|E(D^ns9k|L@b`fHb7(;Tg@h-AcqF(h z7eoRuJ5eBrhp#7!s{T*H|JMJXp`qGD9e-Gd1k|qEM!os~eC<#K>W;zHB2fPv;0^vf z`u~SPzy3x5aF_1{>WaacP>&;AYU-h-F72O9Q!y+&_a_GaSDc5tm4mZgJS0GmvLp~S z)KMC$I8aps3#fqvELKAmg~F+W>Kf`8JRUrOQ2KvE%n>I}uwC`Aa`*6dwpF delta 5774 zcmai$cT|(v*2aN|3esz+MtTu=(-Y9p(NT(m3@r#jsVYhlLBUr-kt!Vp0TBe0Hi95k z7bE}WHe`q zW^UpoocBLisjCLdDUrs`5O=JlJzBknB`Rjm#Vs94Ofr`Gcw)`eWNR=>=ucs&O<*i8 zdZ05DbtJRLVSV*BWHS1aEErYPrv*hnFKCe}>OnY)GrhU}^bausB)neWvxTV*^l3Y^ zXJx2ls;pB2PBt*lyZ<7jQB-dv^8Dr7lrCXe&0H7bW3al*A3FVedtDiSKkxbpv?wQv0xp=*g=%@M9^V*b`V<*A3ol${PvY58C zEq%vN&(_mdcIc6g0L=Q+LZn$24sBv#a*Ba83B;b1~KJ^_LLEMSP8_ z5-aa*M3`8m+v}F+$oi7;F|CF2l5oy?2SP4EEQ?_2E0j0V`^`(Ajb{ed*-|6heFzf* zb22m#XWB1{jwId%AL&GDX?}K;II%58ve_?&jOW9Y-?2T{RCBizV-VM-AMYuVDVK#NbRdSIXl0Y|EkQX|FKzSus`E0nr`@t)Oe&lNGuzUW^HioZ(SPAORw8xz5OsuM}XpS&+x>80`cd%A4=hWslk}H;o@%L98(4MMOPezuee_s8kTN5eQ^& zSjL9Mnh&ZgO*lW0E(My$s)&3lL1|Er@GugVA5tv&{`0ZqTp{&RU2`LfT*TrCbX6Sc-zu$G+i>Oh?tJn$lCq3H}S#dz}|? 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