<s>[INST] Find $\exp\bigg[t\left ( \begin{matrix} -1& 1 \\ 0 & -1 \\ \end{matrix} \right )\bigg]$ by definition Find $\exp\bigg[t\left ( \begin{matrix}
    -1& 1 \\
    0 & -1 \\
   \end{matrix} \right )\bigg]$ by definition
Denote $A=\left ( \begin{matrix}
    -1& 1 \\
    0 & -1 \\
   \end{matrix} \right )$
I find out that
$$A^n=\left ( \begin{matrix}
    -1& n \\
    0 & -1 \\
   \end{matrix} \right ) \forall n\in\mathbb{N} \text{ such that } n\mod 2 =1$$
$$A^n=\left ( \begin{matrix}
    1& -n \\
    0 & 1 \\
   \end{matrix} \right ) \forall n\in\mathbb{N} \text{ such that } n\mod 2 =0$$
Denote $k=2n,j=2n+1$.
$e^{tA}=\sum_{k=1}^\infty \frac{t^k}{k!}A^k + \sum_{j=1}^\infty \frac{t^j}{j!}A^j$
I got a bit confused how to find $e^{tA}$ by definition
Thanks !
 [/INST] From your insight, it may be more helpful to rewrite the powers of $A$ (you made a slight error) as
$$A^n = (-1)^n \begin{pmatrix}
1 & -n \\ 0 & 1 \end{pmatrix}$$
So,
$$\sum_{n=0}^{\infty} \frac{t^n}{n!}A^n = \sum_{n=0}^\infty \frac{(-t)^n}{n!} \begin{pmatrix}
1 & -n \\ 0 & 1 \end{pmatrix} = \sum_{n=0}^\infty \begin{pmatrix}
\frac{(-t)^n}{n!} & -\frac{(-t)^n}{(n-1)!} \\ 0 & \frac{(-t)^n}{n!} \end{pmatrix}$$
Then you can compute these sums component-wise using the Taylor expansion for $e$.
</s>