Commit 19289172 authored by Claude Meny's avatar Claude Meny

Update cheatsheet.fr.md

parent 15993119
Pipeline #16419 canceled with stage
...@@ -70,7 +70,7 @@ $`M^0 = I_m`$ matrice identité de dimension $`m\times m`$. ...@@ -70,7 +70,7 @@ $`M^0 = I_m`$ matrice identité de dimension $`m\times m`$.
*$`\mathbf{M = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}}`$* *$`\mathbf{M = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix}}`$*
**$`\mathbf{e^{\,M}}`$** $`\displaystyle\; = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!}`$ **$`\large{\mathbf{e^{\,M}}}`$** $`\displaystyle\; = \sum_{n=0}^{+\infty}\dfrac{M^n}{n!}`$
<br> <br>
$`\begin{align} \quad\;\; & =\;\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} + $`\begin{align} \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 \\ \dfrac{1}{1!}\;\begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_m \\ \end{pmatrix} + \cdots \\
...@@ -92,10 +92,10 @@ $`\begin{align} \quad\;\; & = \;\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 & ...@@ -92,10 +92,10 @@ $`\begin{align} \quad\;\; & = \;\dfrac{1}{0!}\;\begin{pmatrix} 1 & 0 & 0 \\ 0 &
\end{align}`$ \end{align}`$
<br> <br>
<br> <br>
$`\quad\;\; = \;\begin{pmatrix} \displaystyle\sum_{n=0}^{+\infty}\,\dfrac{\lambda_1^n}{n!} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \displaystyle\sum_{n=0}^{+\infty}\,\dfrac{\lambda_m^n}(n!} \\ \end{pmatrix}`$ $`\quad\;\; = \;\begin{pmatrix} \displaystyle\sum_{n=0}^{+\infty}\,\dfrac{\lambda_1^n}{n!} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \displaystyle\sum_{n=0}^{+\infty}\,\dfrac{\lambda_m^n}{n!} \\ \end{pmatrix}`$
<br> <br>
<br> <br>
**$`\boldsymbol{\mathbf{e^{\,M} = \;\begin{pmatrix} e^{\,\lambda_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & e^{\lambda_m} \\ \end{pmatrix}}}`$** **$`\large{\mathbf{e^{\,M} = \;\begin{pmatrix} e^{\,\lambda_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & e^{\lambda_m} \\ \end{pmatrix}}}`$**
##### $`M`$ est non diagonale, mais diagonalisable : ##### $`M`$ est non diagonale, mais diagonalisable :
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