Undergrad Exponential Operators: Inverting, Rearranging, Expanding

[dyn]

If I rearrange an equation invoving exponentials of operators and I take e^x to the opposite side of the equation it becomes e^-x. What happens if I try to take e^A to the opposite side ? I know a exponential of operators can be expanded as a Taylor series which involves products of matrices but can this be inverted ?

 

[micromass]

Yes, $(e^{A})^{-1} = e^{-A}$

 

[dyn]

Thanks. Are there any conditions for that to apply ? To invert an ordinary matrix requires a non-zero determinant. Are there any conditions on the operator/matrix in the exponential ? Also when taking the exponential over to the other side of the equation I presume order matters in case any operators do not commute ?

 

[micromass]

Thanks. Are there any conditions for that to apply ?

No. As long as $e^A$ exists (which it always does if $A$ is a bounded operator), then the above applies.

Also when taking the exponential over to the other side of the equation I presume order matters in case any operators do not commute ?

Yes.