From 6ab5f133422ba15b94016c92bb48041ce99caef4 Mon Sep 17 00:00:00 2001 From: Feror Date: Mon, 13 Apr 2026 08:46:38 +0200 Subject: [PATCH] Mon Apr 13 08:46:38 CEST 2026 --- Automatisation/TD1.md | 14 ++-- MathématiquesGénérales/CI-SST81E6_TD3.pdf | Bin 0 -> 40712 bytes MathématiquesGénérales/TD2.md | 93 ++++++++++++++++++++++ MathématiquesGénérales/TD3.md | 80 +++++++++++++++++++ MathématiquesGénérales/lib.md | 21 +++++ Matériaux/Index.md | 0 Électrotechnique/Index.md | 0 7 files changed, 201 insertions(+), 7 deletions(-) create mode 100644 MathématiquesGénérales/CI-SST81E6_TD3.pdf create mode 100644 MathématiquesGénérales/TD3.md create mode 100644 MathématiquesGénérales/lib.md create mode 100644 Matériaux/Index.md create mode 100644 Électrotechnique/Index.md diff --git a/Automatisation/TD1.md b/Automatisation/TD1.md index 8a76c6a..02b1efe 100644 --- a/Automatisation/TD1.md +++ b/Automatisation/TD1.md @@ -25,14 +25,14 @@ $H_1$ ```mermaid flowchart LR -A(["E(p)"]) --> B(["+/-"]) --> C("K1(p)") --> D("K2(p)") --> S("S(p)") -D --> E("K3(p)") --> B +A(["$$E(p)$$"]) --> B(["+/-"]) --> C("$$K_1(p)$$") --> D("$$K_2(p)$$") --> S("$$S(p)$$") +D --> E("$$K_3(p)$$") --> B ``` ```mermaid flowchart LR -A(["E(p)"]) --> B(["+/-"]) --> D("K1(p)K2(p)") --> S("S(p)") -D --> E("K3(p)") --> B +A(["$$E(p)$$"]) --> B(["+/-"]) --> D("$$K_1(p)K_2(p)$$") --> S("$$S(p)$$") +D --> E("$$K_3(p)$$") --> B ``` ```mermaid flowchart LR @@ -43,9 +43,9 @@ $H_2$ ```mermaid flowchart LR -A(["E(p)"]) --> B(["+/-"]) --> C("K1(p)") --> B2(["+/-"]) --> D("K2(p)") --> S("S(p)") -C --> E("K4(p)") --> B -D --> E2("K4(p)") --> B2 +A(["$$E(p)$$"]) --> B(["+/-"]) --> C("$$K_1(p)$$") --> B2(["+/-"]) --> D("$$K_2(p)$$") --> S("$$S(p)$$") +C --> E("$$K_4(p)$$") --> B +D --> E2("$$K_4(p)$$") --> B2 ``` ```mermaid flowchart LR diff --git a/MathématiquesGénérales/CI-SST81E6_TD3.pdf b/MathématiquesGénérales/CI-SST81E6_TD3.pdf new file mode 100644 index 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zLcaEb`5t+;VGj7aKY=&j#hW%Zrq5hfNLY+*VGs2^JLE+x@yUQMzUjBgk~gD#H0|R$ zdJ@hM^nu2butdl~&t8EUcxzs6B<-wRo=(}Nt|Y*XzR!hh3;ei=O%(5xqkfQ0NWO@V z@N#^pTYT~0;qwa{Q&zKboBeK;Fvw9Lr^VM|);EHTfx|OAyhU-ZAO@nw=-b7W#E#M( z8RL;aULLirIc|tziBB0+q+z1#LviVc#EHTW53YvW!5W`q?$51Z;fF2ejnA7Z3Xuyj zu<&U~9wvK;$o%1I)W??R)zeAA3*5tN5e>X`^r2aH^V2D9@&N)?Cc}|W>=O=8?1?#b z05=$LM^EwZ@;TejdTR4+^PRyiq=V=50Vtz zPP3l$aaTkgpe$zW#XmZyu@1>7ah5BKyaT-N72=15hvu_C`ekoXK3UA{gKbP_Xfj zK~c*Jm&jx=fV`MaAzNcGIP$VOB{hUKMau_6u2G)TDK$7*QUjq6QvyTuRP{mnDmqXd zsIH!>sxmn{FBpMiKl zUTLY{8V~Bu$c9A-D+28h)?9Xm`nh8nUE0P5zz|}(wfK?Jm$ZURmp!4F{JWprz{Zvt zSe$w&=O)iTtVIWdrOPBT9QgS?-<;M_#AInI;4irCM2 z08Y|7?Zw2%Kg5J6`GNm^2WtcT9xKFMgp&gLK6w9N`=JGr;n65BVb(c%U1NR6q34eD z+xLWmP6zI0}$ + + +## 4) +Soit $t\in\mathbb{R}$ +$e^{tA}=\sum_{k=0}^{\infty}\frac{(tA)^k}{k!}$ +$=\sum_{k=0}^{\infty}(\frac{t^k}{k!}(-1)^{k+1})A+I$ +$=-A(\sum_{k=0}^{\infty}\frac{(-t)^k}{k!}-1)+I$ +$=-Ae^{-t}+A+I$ +$=P_0e^{\lambda_0t}+P_1e^{\lambda_1t}$ + +$P_0=-A=\pmatrix{1&0&-1\\-1&0&1\\0&0&0}$ +$P_1=A+I=\pmatrix{0&0&1\\1&1&-1\\0&0&1}$ +Vecteur propre dans $P_0$: $\pmatrix{1\\-1\\0}$ +Vecteurs propres dans $P_1$: $\pmatrix{0\\1\\0} \text{ et } \pmatrix{1\\-1\\1}$ + +# Exercice 2: +## 1. +$B=\pmatrix{a&b&c\\a&b&c\\a&b&c}\,(a,b,c)\in\mathbb{R}^3$. + +$\mathcal{X}_B(x)=\det{(xI-B)}=\begin{vmatrix}x-a&-b&-c\\-a&x-b&-c\\-a&-b&x-c\end{vmatrix}=\begin{vmatrix}x-(a+b+c)&-b&-c\\x-(a+b+c)&x-b&-c\\x-(a+b+c)&-b&x-c\end{vmatrix}$ + +$\mathcal{X}_B(x)=(x-(a+b+c))\begin{vmatrix}1&-b&-c\\1&x-b&-c\\1&-b&x-c\end{vmatrix}\matrix{\\L_2-L_1\\L_3-L_1}$ + +$\mathcal{X}_B(x)=(x-(a+b+c))\begin{vmatrix}1&-b&-c\\0&x&0\\0&0&x\end{vmatrix}=x^2(x-(a+b+c))$ + +On pose $p:x\rightarrow x(x-(a+b+c))$ +- [x] Divise $\mathcal{X}_B$ +- [x] Mêmes racines que $\mathcal{X}_B$ +- [x] Unitaire +- [x] De + bas degré possible + +$p(B)=B(B-(a+b+c)I)$ +$p(B)=\pmatrix{a&b&c\\a&b&c\\a&b&c}\pmatrix{-(b+c)&b&c\\a&-(a+c)&c\\a&b&-(a+b)}=0$ + +$p$ est le polynome minimal de $B$. +$S_p(B)=\{0,a+b+c\}$. +$p$ est scindé à racines simples si $a+b+c≠0$ + +La matrice $B$ est donc diagonalisable $\forall(a,b,c)\in\mathbb{R}^3$ tel que $a+b+c≠0$ + +## 2. +Soit $\lambda_0=0$ et $\lambda_1=a+b+c$. +$\cases{\lambda_0;E_0;P_0\\\lambda_1;E_1;P_1}$ +$\forall n\in\mathbb{N},\,\lambda_0^nP_0+\lambda_1^nP_1=B^n$ + +$\cases{P_0+P_1=I\\(a+b+c)P_1=B}\implies\cases{P_1=\frac{1}{a+b+c}B\\P_0=I-\frac{1}{a+b+c}B=\frac{-1}{a+b+c}(B-(a+b+c)I)}$ + +$B^n=\cases{I \text{ si }n=0\\(a+b+c)^nP_1=(a+b+c)^{n-1}B \text{ si }n>0}$ + +$e^{tB}=e^{0t}P_0+e^{(a+b+c)t}P_1$. + +Vecteurs propres de l'espace $E_0$: $\pmatrix{-(b+c)\\a\\a}$ et $\pmatrix{b\\-(a+c)\\b}$ +Vecteur propre de l'espace $E_1$: $\pmatrix{a\\a\\a}$ + +# Exercice 3: + +$C=\pmatrix{3&1&0&1\\1&3&0&1\\0&0&4&0\\0&0&0&2}$ + +$\mathcal{X}_C(x)=\begin{vmatrix}x-3&-1&0&-1\\-1&x-3&0&-1\\0&0&x-4&0\\0&0&0&x-2\end{vmatrix}=(x-2)(x-4)((x-3)^2-1)=(x-2)^2(x-4)^2$ + +On pose $p: x\rightarrow(x-2)(x-4)$ +- [x] Divise $\mathcal{X}_B$ +- [x] Mêmes racines que $\mathcal{X}_B$ +- [x] Unitaire +- [x] De + bas degré possible + +$p(C)=(C-2I)(C-4I)=0_4$. + +$p$ est le polynome minimal de $C$, il est scindé et à racines simples. Donc, $C$ est diagonalisable. + +$S_p(C)=\{2,4\}$. + +$\forall n\in\mathbb{N},\,\lambda_0^nP_0+\lambda_1^nP_1=C^n$ +$\cases{P_0+P_1=I\\2P_0+4P_1=C}\implies\cases{P_0=I-P_1\\2I-2P_1=C}$ +$P_1=\frac{C-2I}{2}$ +$P_0=I-\frac{C-2I}{2}=-\frac{1}{2}(C-4I)$. + +$C^n=2^nP_0+4^nP_1$ + +$e^{tC}=e^{2t}P_0+e^{4t}P_1$ \ No newline at end of file diff --git a/MathématiquesGénérales/TD3.md b/MathématiquesGénérales/TD3.md new file mode 100644 index 0000000..5b2d5a8 --- /dev/null +++ b/MathématiquesGénérales/TD3.md @@ -0,0 +1,80 @@ +![[CI-SST81E6_TD3.pdf]] +![[lib]] + +# Exercice 2: +On pose $A=\pmatrix{0&0&0\\1&1&0\\1&0&0}$. +## 1. +$\det{(xI-A)}=\det\begin{vmatrix}x&0&0\\-1&x-1&0\\-1&0&x\end{vmatrix}$ +$\det{(xI-A)}=x^2(x-1)$ + +On cherche le polynome minimal: +On pose $P:x\rightarrow x(x-1)$ +$\mathcal{X}_A(A)=0=A^2(A-I)=A^3-A^2=0$ +Donc $A^3=A^2$. + +## 2. +$P(A)=A(A-I)=\pmatrix{0&0&0\\1&1&0\\1&0&0}\pmatrix{-1&0&0\\1&0&0\\1&0&-1}=\pmatrix{0&0&0\\0&0&0\\-1&0&0}≠0$ +Donc $P$ n'est pas le polynome minimal de $A$. + +Le polynome minimal est à racine multiple, donc $A$ n'est pas diagonalisable. + +## 3. +Spectre de $A$: +$S_p(A)=\{0;1\}$. +$\lambda_0=0;\lambda_1=1;$ +$\dim(E_0)=1$ +$\dim(E_1)=1$ + +## 4. +$A=D+N$ +Avec $D$ "Diagonalisable" et $N$ "Nilpotente". $DN=ND$. +### a) +$D=q(A)$ avec $q\in\mathbb{R}(x)$ tel que $\cases{q\equiv\lambda_i[(x-\lambda_i)^{\mu_i}]\\\mu_i=\text{ Multiplicité de }\lambda_i=2}$ +$\cases{q\equiv0[x^2] \\ q\equiv1[x-1]}$ +Les deux modulis sont premiers entre eux car ils n'ont pas de racines communes. +$\exists (u,v)\in\mathbb{R}(x)$ tel que $\cases{q=0+ux^2\\q=1+v(x-1)}$ +Euclide: +$\div{x^2}{x-1}{x+1}{-(x^2-x)\\+x\\-(x-1)\\1}$ + +### b) +$D_0=A$ +$D_{n+1}=D_n-P(D_n)\times(P'(D_n))^{-1}$ +$D=D_n$ dès que $2^n≥\max{(\mu_i)}$ +Avec $P\in\R(x), P=\frac{\mathcal{X}_A}{PGDC(\mathcal{X}_A,\mathcal{X}_A')}$ + +$\mathcal{X}_A=x^2(x-1)=x^3-x^2$ +$\mathcal{X}_A'=3x^2-2x=x(3x-2)$ +$PGDC(\mathcal{X}_A,\mathcal{X}_A')=x$ + +$D=D_1=D_0-P(D_0)\times(P'(D_0))^{-1}=A-P(A)\times(P'(A))^{-1}$ Avec $P'=2x-1$ +$P'(A)=2A-I=\pmatrix{-1&0&0\\2&1&0\\2&0&-1}$ +$(P'(A))^{-1}=\pmatrix{-1&0&0\\2&1&0\\2&0&-1}^{-1}=\pmatrix{\frac{1}{-1}&0&0\\?&\frac{1}{1}&0\\?&?&\frac{1}{-1}}$ +$D=\pmatrix{0&0&0\\1&1&0\\1&0&0}-\pmatrix{0&0&0\\0&0&0\\-1&0&0}\pmatrix{\frac{1}{-1}&0&0\\?&\frac{1}{1}&0\\?&?&\frac{1}{-1}}=\pmatrix{0&0&0\\0&0&0\\1&0&0}$ +## 5. +$D^n=D$: Initialisation avec $n=1$ car $\N^*$ +$D^1=D$ +Hérédité: Soit $n\in\N^*$ on suppose $D^n=D$ +Montrons que $D^{n+1}=D$ +$D^{n+1}=D^n\times D=D^2$ +$D^{n+1}=A^4$ +$D^{n+1}=A\times A^3$ +$D^{n+1}=A\times A^2$ +$D^{n+1}=A^3$ +$D^{n+1}=A^2$ +$D^{n+1}=D$ + +Conclusion: $\forall n\in\N^*;D^n=D$ +$$ +A^n=(D+N)^n=\sum_{k=0}^n\pmatrix{n\\k}D^{n-k}N^k +$$ +$$ +A^n=\cases{ +n=0\implies I \\ +n=1\implies A \\ +n≥2\implies D^n=A^2 +} +$$ + +$$e^{tD}=\sum_{k=0}^{\infty}\frac{(tD)^k}{k!}=(\sum_{k=1}^{\infty}\frac{t^k}{k!})\times D+I=(\sum_{k=0}^{\infty}\frac{t^k}{k!} -1)\times D + I=(e^t-1)D+I$$ +$$e^{tA}=\sum_{k=0}^{\infty}\frac{t^k}{k!}A^k=(\sum_{k=0}^{\infty}\frac{t^k}{k!}-1-t)A^2+I+tA=D(e^t-1)+I+t(A-D)$$ + diff --git a/MathématiquesGénérales/lib.md b/MathématiquesGénérales/lib.md new file mode 100644 index 0000000..996fa7d --- /dev/null +++ b/MathématiquesGénérales/lib.md @@ -0,0 +1,21 @@ +$$ +\newcommand{\div}[4]{ +\begin{array}{r|l} +#1 & #2\\ +\hline +\begin{array}[t]{r}#4\end{array} & \begin{array}[t]{l}#3\end{array} +\end{array} +} + +\newcommand{\matp}[3]{ +\begin{array}{cc} +& #2 \\ +#1 & #3 +\end{array} +} + +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Z}{\mathbb{Z}} +$$ \ No newline at end of file diff --git a/Matériaux/Index.md b/Matériaux/Index.md new file mode 100644 index 0000000..e69de29 diff --git a/Électrotechnique/Index.md b/Électrotechnique/Index.md new file mode 100644 index 0000000..e69de29