We just saw today the concept of the exponential of a matrix. Let $A\in \mathbb R^{n\times n}$ a matrix. How can I prove that $$e^A:=\sum_{k=0}^\infty \frac{A^k}{k!}$$ is well defined ?
What I've done so far
Case 1: If $D$ is diagonal, then $$(e^{D})_{ij}=\begin{cases}e^{d_i}&i=j\\0&otherwise\end{cases}.$$ Therefore, it's well defined (i.e. that $e^1\in \mathbb R^{n\times n}$).
Case 2: If $A$ is diagonalizable, then $A=\Lambda D\Lambda ^{-1}$ for some invertible matrix $\Lambda $ and some diagonal matrix $D$. Then I could prove that $e^A=\Lambda e^D\Lambda ^{-1}$, and thus, $e^A$ is well defined.
Case 3: If $A$ is triangular, then $A=D+N$ for some diagonal matrix $D$ and some nilpotent matrix $N$. Since $A^k=D^k$ for all $k\geq n$, we are in the same case than the Case 1.
Case 4: If $A$ is trigonalizable, combining the Cases 2 and 3 gives the wished result.
Question How can I do in the case where $A$ is not trigonalisable ?