A problem on matrix exponentials: ${{e}^{AB}}$ and ${{({{e}^{A}})}^{B}}$

[guarcien]

Does anyone know if there is a link between these two expressions,below, where A and B are square and commutable matrices?

${{e}^{AB}}$ and ${{({{e}^{A}})}^{B}}$

 

[topsquark]

Does anyone know if there is a link between these two expressions,below, where A and B are square and commutable matrices?

${{e}^{AB}}$ and ${{({{e}^{A}})}^{B}}$

The meaning of the form $e^{X}$, where X is a square matrix, is given by the Taylor expansion of the exponential, similar to the series for $e^x$:
$\displaystyle e^{X} \equiv \sum_{n = 0}^{ \infty } \dfrac{1}{n!} X^n$

In order to discuss $\left ( e^A \right ) ^B$ we will need a Taylor series to represent this. But this approach is severely compromised by the fact that A and B may not commute. For example, in general $e^A e^B \neq e^{A + B}$ unless A and B commute. I know of no way to approach this problem.

-Dan