# Math Help - Functions of Matrices

1. ## Functions of Matrices

Let $D: \mathbb {R} [X] \to \mathbb {R} [X]$ be the differentiation operator $D(f(X)) = f'(X)$. Prove that $e^{tD} (f(X)) = f(X+t)$ for a real number $t \in \mathbb {R}$.

I really don't know where to start on this and would appreciate if anyone could give me hints on what actually needs to be shown in a more concrete way. Anything that could start me off would be great! thanks

2. Hint :

$e^{tD}=1+tD+\frac{t^2D^2}{2!}+...$

$e^{tD}f(x)=1+tf'(x)+\frac{t^2f''(x)}{2!}+...$

Now think in terms of Taylor series...

3. Originally Posted by Bruno J.

$e^{tD}f(x)=1+tf'(x)+\frac{t^2f''(x)}{2!}+...$
that $1$ should actually be $f(x).$

4. Indeed Thanks!