Is there any way to reformulate exponentials of matrix products ($e^{XY}$), where $X,Y\in \mathbb R^{n\times n}$? I am interested in how $e^X, e^Y$ relate to $e^{XY}$. Of particular interest to my research question are:
- Exponentials of matrix exponents $e^{X^k}$
- Exponentials of matrix cross-products $e^{X^TX}$
I understand that with scalars $x,y\in\mathbb R$ we have the identity $e^{xy}=(e^x)^y$, and I wonder if an analogous identity is available for matrices. I am familiar with the rule that, where $Y^{-1}$ exists, we have: $$e^{YXY^{-1}}=Ye^XY^{-1}$$ However I cannot find any further information on more general products.