Commit 525239ff authored by Claude Meny's avatar Claude Meny

Update cheatsheet.fr.md

parent 5f9c8262
Pipeline #16464 canceled with stage
......@@ -154,15 +154,15 @@ $`\quad\;\; = \;\begin{pmatrix} \displaystyle\sum_{n=0}^{+\infty}\,\dfrac{\lambd
avec les lmbdas valeurs propres et P ... à terminer, en fonction de ce qu'on met avant.
**$`\mathbf{e^{\,M}`$**$`\;=\displaystyle\sum_{k=0}^{\infty}\dfrac{M^k}{k!}`$
**$`\large\mathbf{e^{\,M}}}`$**$`\;=\displaystyle\sum_{k=0}^{\infty}\dfrac{M^k}{k!}`$
$`\quad\;=\displaystyle\sum_{k=0}^{\infty}\left(\dfrac{1}{k!}\,P\,\begin{pmatrix}\lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n^k\\ \end{pmatrix}\,P^{\,-1}\right)`$
$`\quad\;=\displaystyle\sum_{k=0}^{\infty}\dfrac{1}{k!}\,P\,\begin{pmatrix}\lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n^k\\ \end{pmatrix}\,P^{\,-1}`$
$`\quad\;=\displaystyle P\left(\,\sum_{k=0}^{\infty}\dfrac{1}{k!}\,\begin{pmatrix}\lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n^k\\ \end{pmatrix}\right)\,P^{\,-1}`$
$`\quad\;=\displaystyle P\left[\,\sum_{k=0}^{\infty}\dfrac{1}{k!}\,\begin{pmatrix}\lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n^k\\ \end{pmatrix}\;\right]\,P^{\,-1}`$
$`\quad\;=\displaystyle P\left(\begin{pmatrix}\sum_{k=0}^{\infty}\dfrac{1}{k!}\lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \sum_{k=0}^{\infty}\dfrac{1}{k!}\lambda_n^k\\ \end{pmatrix}\right)\,P^{\,-1}`$
$`\quad\;=\displaystyle P\;\begin{pmatrix}\sum_{k=0}^{\infty}\dfrac{1}{k!}\lambda_1^k & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \sum_{k=0}^{\infty}\dfrac{1}{k!}\lambda_n^k\\ \end{pmatrix}\;P^{\,-1}`$
**$`\large{\mathbf{e^{\,M}=\displaystyle P\left(\begin{pmatrix} e^{\,\lambda_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 &e^{\,\lambda_n}\\ \end{pmatrix}\right)\,P^{\,-1}}`$**
**$`\large{\mathbf{e^{\,M}=\displaystyle P\;\begin{pmatrix} e^{\,\lambda_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 &e^{\,\lambda_n}\\ \end{pmatrix}\;P^{\,-1}}}`$**
......@@ -170,17 +170,17 @@ $`\quad\;=\displaystyle P\left(\begin{pmatrix}\sum_{k=0}^{\infty}\dfrac{1}{k!}\l
Elles sont analogues aux propriétés de la fonction exponentielle d'un nombre réel :
* *$`e^{\,0} = I_n`$*,
* **$`\large{\mathbf{e^{\,0} = I_n}}`$**,
avec $`I_n`$ matrice carré identité de dimensions $`n\times n`$
* Si $`A`$ et $`B`$ commutent $`(AB = BA)`$, alors :
*$`e^{\,A+B} = e^A\,e^B = e^{\,B+A} = e^B\,e^A `$*
*$`\large{\mathbf{e^{\,A+B} = e^A\,e^B = e^{\,B+A} = e^B\,e^A}} `$*
* *$`\big(e^A\big)^{\,-1} = e^{- A}`$*
* *$`\large{\mathbf{\big(e^A\big)^{-1} = e^{- A}}}`$*
* *$`e^{A}\,A = A\,e^{A\} `$*
* *$`\large{\mathbf{e^{\,A}\,A = A\,e^{A\}}} `$*
* *$`\dfrac{d}{dt}e^{A t} = A\,e^{A t}`$*
* *$`\large{\mathbf{\dfrac{d}{dt}e^{A t} = A\,e^{A t}}}`$*
......
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