From 3a1e8f3d0758c2b7eb1bc3b13fc3bccff2c531ba Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 29 Mar 2023 19:24:51 +0200 Subject: [PATCH] feat(geometria): aggiorna gli appunti del 29/03/2023 --- .../main.pdf | Bin 151972 -> 153200 bytes .../main.tex | 12 +++++++++--- 2 files changed, 9 insertions(+), 3 deletions(-) rename Geometria I/Teoria spettrale degli endomorfismi/{2023-03-29 => 2023-03-29, Es. Computo delle basi di Jordan}/main.pdf (75%) rename Geometria I/Teoria spettrale degli endomorfismi/{2023-03-29 => 2023-03-29, Es. Computo delle basi di Jordan}/main.tex (79%) diff --git a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-29/main.pdf b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.pdf similarity index 75% rename from Geometria I/Teoria spettrale degli endomorfismi/2023-03-29/main.pdf rename to Geometria I/Teoria spettrale degli endomorfismi/2023-03-29, Es. 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Computo delle basi di Jordan/main.tex similarity index 79% rename from Geometria I/Teoria spettrale degli endomorfismi/2023-03-29/main.tex rename to Geometria I/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.tex index 6e18020..32a9987 100644 --- a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-29/main.tex +++ b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.tex @@ -65,7 +65,7 @@ da cui si ottiene anche che $p_A(t) = (t-2)^2 (t-2)$. Si calcola ora una base di Jordan per $A$. \\ - Sia $\Ker (A - I)^2 = \Ker (A - I) \oplus U_1$. $\dim U_1 = 1$, + ($\lambda = 1$) Sia $\Ker (A - I)^2 = \Ker (A - I) \oplus U_1$. $\dim U_1 = 1$, e poiché $\e5 \in \Ker (A - I)^2$, ma $\e5 \notin \Ker (A-I)$, vale che $U_1 = \Span(\e5)$. \\ @@ -74,7 +74,7 @@ non appartiene a $\Span(A \e5)$, si ottiene che una base relativa al blocco di $1$ è $A \e5, \e5, \e5+\e1-\e3$. - Per quanto riguarda invee il blocco relativo a $2$, essendo + ($\lambda = 2$) Per quanto riguarda invece il blocco relativo a $2$, essendo tale blocco diagonale, è sufficiente ricavare una base di $\Ker (A-2I)$, come $\e4$ e $\e1 + \e3$. \end{example} @@ -93,7 +93,13 @@ \end{proposition} \begin{proof} - Si osserva che $J_{0,m} B = \Matrix{B_2 \\ \hline B_3 \\ \hline \vdots \\ \hline B_{m} \\ \hline 0}$, mentre $B J_{0,m} = \Matrix{0 & \rvline & B^1 & \rvline & B^2 & \rvline & \cdots & \rvline & B^{m-1} }$. + Sia $B \in C(J_{0, m})$. Si osserva che $J_{0,m} B = \Matrix{B_2 \\ \hline B_3 \\ \hline \vdots \\ \hline B_{m} \\ \hline 0}$, mentre $B J_{0,m} = \Matrix{0 & \rvline & B^1 & \rvline & B^2 & \rvline & \cdots & \rvline & B^{m-1} }$. Per ipotesi deve valere che $J_{0,m} B = B J_{0,m}$, e quindi, uguagliando le matrici colonna a colonna, si osserva + la colonna $B^1$ è tutta nulla eccetto per il primo elemento; si osserva poi che la colonna $B^2$ è composta + da elementi di $B^1$ traslata in basso di una posizione; e così via ciclando sulle colonne, ottenendo che, + data $B^m = \Vector{ a_0 \\ a_1 \\ \vdots \\ a_{m-1} }$, $B = a_0 I + a_1 J_{0,m} + \ldots + a_{m-1} J_{0, m}^{m-1}$, + quindi $B \in \Span(I, J_{0, m}, J_{0, m}^2, ..., J_{0, m}^{m-1})$. Dal momento che ogni elemento generatore di + $\Span(I, J_{0, m}, J_{0, m}^2, ..., J_{0, m}^{m-1})$ commuta con $J_{0,m}$, vale la doppia inclusione, da cui + la tesi. \end{proof} \begin{remark}