What are the possible $n \times n$ matrices such that $e^X = I$? Some obvious ones are $X = 2\pi i m I$ for integer $m$. Are there others and do they have some kind of group structure?
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1A simple extension would be any diagonalizable matrix with eigenvalues taking values in ${2\pi i m| m\in\mathbb{Z}}$, which is not closed under addition or multiplication. – whpowell96 Dec 12 '23 at 17:50
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Hints to get you started: What values might you put on the diagonal of a diagonal matrix $X$? $e^Xe^Y = e^{X+Y}$ if $X$ and $Y$ commute. – Ethan Bolker Dec 12 '23 at 17:51
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And matrices similar to the ones in the @whpowell96 comment. Are there others? Investigate the Jordan form. – GEdgar Dec 12 '23 at 17:51