From e1949180f3b091ff0bb5876c9fffc9398f168433 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 5 May 2023 22:15:49 +0200 Subject: [PATCH] feat(geometria): aggiunge segnatura dei prodotti (semi)definiti --- .../1. Introduzione al prodotto scalare.tex | 10 ++++++++++ .../main.pdf | Bin 432788 -> 433796 bytes 2 files changed, 10 insertions(+) diff --git a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex index 3373d2e..d2ba1df 100644 --- a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex +++ b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex @@ -597,6 +597,16 @@ stessi numeri $\iota_+$, $\iota_-$ e $\iota_0$ di vettori di base con forma quadratica positiva, negativa e nulla. \\ + \li Vale che $\varphi$ è definito positivo $\iff$ $\sigma = (n, 0, 0)$. Infatti, per il teorema + di Sylvester reale, $i_+ = n$ $\iff$ la dimensione del massimo sottospazio di $V$ su cui $\varphi$ + è definito positivo è $n$ $\iff$ $\varphi$ è definito positivo. Analogamente $\varphi$ è definito + negativo $\iff$ $\sigma = (0, n, 0)$. \\ + + \li Nello stesso spirito dei prodotti definiti, $\varphi$ è semidefinito positivo $\iff$ $\iota_- = 0$. + Infatti valgono le seguenti equivalenze: $\varphi$ è semidefinito positivo $\iff$ non esiste un vettore + $\v \in V$, $\v \neq \vec 0$ tale che $q(\v) < 0$ $\iff$ $\iota_- = 0$. Analogamente $\varphi$ è semidefinito + negativo $\iff$ $\iota_+ = 0$. \\ + \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. \\ diff --git a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf index f06454eadbf6aab648ccb019d1c68072aacf7e5f..01706176e13074b26d7acb3ef3666ea9eb2c3a9b 100644 GIT binary patch delta 23241 zcmV)yK$5?dwHk!I8i2F`B_$YBW3EhjLpUm#nCk z|9JNw-~CC5{+cVsHDPM85{z&w!kQobJcoL7+I(IrU6e1EE6GLifH3?5%StMXeYL?0 zRn*nPVZV+y)wbGI0oF|zS8vKV?sTk9r?Pong{S-Kv22!!Q-ptm#U8hDh4c2b*$wCR zQ=YefE8OCO!mb2S(uLh@OR`*PZHu@5SGK4R9xnhO{5fyxhcaMB(>`XbCe%on%un(| z90JrK&QL$X5b4Z95U1iwG>2hBxYU5(ZWy2|0CL`!;o^N+S2-B8@xMx8NT+9vIBA<9 zr*d0`DFY3wW4L=bZ#)fkV2QYxG6|rDuy(C~Pr0=e;hHxiex{1%rjfAUcF^ZI!uHSx 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