As I know, exponential function is strictly monotonic increasing function. So it means that for $x,y \in \mathbb{R}$, if $e^x=e^y$ then $x=y$. But how about exponential matrix? If I have two matrices $X,Y$ satisfy $e^X=e^Y$, does that mean $X=Y$?
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peek-a-boo
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by the way, I edited the title to properly reflect what you asked in the question (unique isn't the word you were looking for, the correct term in this context is injective) – peek-a-boo Dec 30 '19 at 13:51
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Ahh thanks @peek-a-boo. – RANGGAJAYA CIPTAWAN Dec 30 '19 at 13:53
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No it does not. Check out the zero matrix and the diagonal matrix $\operatorname{diag}(2i\pi, 0)$ – Cameron Williams Dec 30 '19 at 14:00
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also, you should specify whether you're interested in matrices with only real entries, or whether you allow for complex entries – peek-a-boo Dec 30 '19 at 14:03