Update cheatsheet.fr.md

12.temporary_ins/04-math-tools/70.systems-of-differential-equations/40.parallel-1/cheatsheet.fr.md

@@ -54,47 +54,52 @@ valeurs et vecteurs propres
 
 Soit $`M`$ une matrice réelle carré de dimension $`m\times m`$.
 
-Par définition : $`\forall k\in \mathbb{N}^{\*}\,,\,`$ **$`\mathbf{M^k = \underbrace{M \times M \times \cdots \times M}_{\color{blue}{\text{k fois}}}`$**
+Par définition : $`\forall k\in \mathbb{N}^{*}\,,\,`$ 
+**$`\mathbf{M^k = \underbrace{M \times M \times \cdots \times M}_{\color{blue}{\text{k fois}}}}`$**
 
 ##### $`M`$ est diagonale`$
 
+<br>
+
 *$`\mathbf{M^2}`$* $`\; = M\times M`$
 
-$`\quad \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}\;\times\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}`$
+$`\quad = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}\;\times\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}`$
 
-*$`\mathbf{\quad \begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^2\\ \end{pmatrix}}`$*
+*$`\mathbf{\quad = \begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^2\\ \end{pmatrix}}`$*
 
 Et par récurrence,
 
-$`\forall k \in \mathbb{N}^{\*}\setminus \{1}`$
+$`\forall k \in \mathbb{N}^{*}\{1}`$  <!--\setminus -->
 
 **$`\mathbf{M^k}`$** $`\; = M^{k-1}\times M`$
 
 $`\quad = \begin{pmatrix} \lambda_1^{k-1} & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^{k-1} \\ \end{pmatrix}\;\times\; \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}`$
 
-**$`\mathbf{\quad = \begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^k\\ \end{pmatrix}}`$
+**$`\mathbf{\quad = \begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^k\\ \end{pmatrix}}`$**
 
 
 ##### $`M`$ est non diagonale, mais diagonalisable`$
 
+<br>
+
 *$`\mathbf{M^2}`$* $`\; = M\times M`$
 
-$`\quad P\,\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}\,\underbrace{P^{-1}\;\times\; 
+$`\quad = P\,\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}\,\underbrace{P^{-1}\;\times\; 
 \; P}_{\color{blue}{= I_m}}\,\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}\,P^{-1}`$
 
-$`\quad P\,\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}^{\,2}\,\underbrace{P^{-1}`$
+$`\quad = P\,\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^{\,2}`$
 
-*$`\mathbf{\quad P\,\begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^2\\ \end{pmatrix}\,\underbrace{P^{-1}}`$*
+*$`\mathbf{\quad = P\,\begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^2\\ \end{pmatrix}\,\underbrace{P^{-1}}`$*
 
 Et par récurrence :   
 
-$`\forall k \in \mathbb{N}^{\*}\setminus \{1}`$
+$`\forall k \in \mathbb{N}^{\*}\ {1}`$
 
 **$`\mathbf{M^k}`$** $`\; = M^{k-1}\times M`$
 
 $`\quad = P\,\begin{pmatrix} \lambda_1^{k-1} & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^{k-1} \\ \end{pmatrix}\,\underbrace{P^{-1}\;\times\;  P}_{\color{blue}{= I_m}}\,\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m\\ \end{pmatrix}\,P^{-1}`$
 
-**$`\mathbf{\quad P\,\begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^k\\ \end{pmatrix}\,\underbrace{P^{-1}}`$**
+**$`\mathbf{\quad = P\,\begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \cdots & 0 \\ 0 & 0 & \lambda_m^k\\ \end{pmatrix}\,\underbrace{P^{-1}}`$**
 
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