 Solution

Induction shows $(PA{P}^{-1}{)}^k=P{A}^k{P}^{-1}$ for every nonnegative integer $k$: in a product, adjacent factors $P^{-1}P$ cancel. At $k=0$, this is $I=PI{P}^{-1}$. Hence each partial sum of the matrix exponential satisfiesThe series is absolutely convergent, since a submultiplicative matrix norm bounds it by ${\sum }_k\Vert A{\Vert }^k/k!=e^{\Vert A\Vert }$. Passing to the limit therefore proves that the matrix exponential respects similarity:No diagonalizability assumption is required.