Given real square matrices $A_1,A_2$ of the same dimension, does the following inequality hold?
$e^{A^{T}_{1}}e^{A^{T}_{2}}e^{A_{2}}e^{A_{1}}\leq{e^{A_{1}+A_{2}+A^{T}_{1}+A^{T}_{2}}}$
where $A^{T}_{i}$ stands for the transpose of $A_{i}$ and $\leq$ denotes the negative semi-definiteness of the left argument with respect to right. I know if $A_{1},A_{2}$ commute and both matrices are normal, then
$e^{A^{T}_{1}}e^{A^{T}_{2}}e^{A_{2}}e^{A_{1}}={e^{A_{1}+A_{2}+A^{T}_{1}+A^{T}_{2}}}$. But I am not sure if the inequality mentioned above holds.
Any suggestions, hints or references relating to the results on similar matrix exponential inequalities are greatly appreciated.
It's false. Take $A_1=0_2,A_2=\begin{pmatrix}-7.2&-3.2\\-0.2&-7.4\end{pmatrix}$.