TITLE: Doubt in the defn of exponential operator.
QUESTION [5 upvotes]: definition
Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$,
$$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$
But in this definition, What they are meaning by the term $A^kt^k$, If I give this matrix $$A=\begin{pmatrix}1&-1\\2&3\end{pmatrix},$$
what can we say about the $(1)$, means  How it will be?

REPLY [3 votes]: One way to handle this kind of problem is to find the Jordan Normal Form of $A$:
$$
A=\begin{bmatrix}
-1-i&-1+i\\
2&2
\end{bmatrix}
\begin{bmatrix}
2-i&0\\
0&2+i
\end{bmatrix}
\begin{bmatrix}
-1-i&-1+i\\
2&2
\end{bmatrix}^{-1}
$$
Taking the exponential of $A$ is now pretty simple:
$$
\begin{align}
\exp(At)&=
\begin{bmatrix}
-1-i&-1+i\\
2&2
\end{bmatrix}
\begin{bmatrix}
e^{2t}e^{-it}&0\\
0&e^{2t}e^{it}
\end{bmatrix}
\begin{bmatrix}
-1-i&-1+i\\
2&2
\end{bmatrix}^{-1}\\
&=\begin{bmatrix}
-1-i&-1+i\\
2&2
\end{bmatrix}
\begin{bmatrix}
e^{2t}e^{-it}&0\\
0&e^{2t}e^{it}
\end{bmatrix}
\begin{bmatrix}
2i&1+i\\
-2i&1-i
\end{bmatrix}\\
&=e^{2t}\begin{bmatrix}
\cos(t)-\sin(t)&-\sin(t)\\
2\sin(t)&\cos(t)+\sin(t)
\end{bmatrix}
\end{align}
$$