In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$.
Is that true the other way around? My goal is to calculate all solutions of the equation $e^X = I_n$.
I am not comfortable with using the logarithm on $e^X$ because I know the exponential is not even surjective on $M_n(\Bbb C)$. Any suggestions to tackle this questions?
Thank you for your help.