Prove that $e^{\lambda A}Be^{-\lambda A}=B$

Question

Prove that $$e^{\lambda A}Be^{-\lambda A}=B$$ if $[A,B]=0$. $A$ and $B$ are operators and $\lambda$ is a complex number.

Can anyone explain how I should go about this question? How do I calculate the exponential of an operator via matrix operations? I just need a hint. Any help would be appreciated thanks.

Answer 1

The exponential of a matrix is defined through its power series: $\exp(A)=\sum_{i=0}^\infty \frac{A ^i}{i!}$. Clearly you would like to simplify the two exponentials, but in order to do so you need that...