Update cheatsheet.fr.md

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

@@ -66,22 +66,35 @@ $`e^{\,M}`$ matrice de dimension $`m\times m`$.
 
 $`M^0 = I_m`$ matrice identité de dimension $`m\times m`$.
 
-$`M`$ est diagonale :
+##### $`M`$ est diagonale :
 
-$`M = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}`$
+*$`\mathbf{M = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}}`$*
 
-$`e^{\,M} & = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!}`$   
+**$`\mathbf{e^{\,M}}`$** $`\; = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!}`$   
 <br>
-$`\begin{align} \quad\quad & = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!} \\
+$`\begin{align} \quad\quad & = +\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
++\dfrac{1}{1!}\;\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
 & \\
-& = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
-\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
+& \quad\quad \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^2
++\cdots \\
+& \\
+& \quad\quad \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^k + \cdots
+\end{align}`$    
+<br>
+<br>
+$`\begin{align} \quad\quad & = +\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + 
++\dfrac{1}{1!}\;\begin{pmatrix}  \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
 & \\
-& \quad\quad\quad \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^2
+& \quad\quad \cdots +\dfrac{1}{2!}\;\begin{pmatrix} \lambda_1^2 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m^2 \\ \end{pmatrix}
 +\cdots \\
 & \\
-& \quad\quad\quad \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}^k + \cdots
-\end{align}`$
+& \quad\quad \cdots + \dfrac{1}{k!}\;\begin{pmatrix} \lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m^k \\ \end{pmatrix} + \cdots
+\end{align}`$    
+<br>
+
+
+
+
 
 $`\begin{align}\quad\quad\quad & = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!} \\
 & \\