# When is the exponential map between matrices injective?

Consider a complex matrix $A$. When does $e^A=e^B$ imply $A=B$?

Is there any general statement that can be made as to when this holds?

It is clearly not true in general, a trivial example being when $A$ and $B$ are diagonal in the same basis but with eigenvalues differing by $2\pi i\mathbb Z$. This is also mentioned in this question.

This answer also seems to provide an answer to this question, but I'm not familiar with the theory of Lie algebras so I'm not sure, and I wouldn't know how to translate it into more elementary statements about matrices (if that is even possible).

• The answer of Yves does really provide an answer, in my opinion. This question should be considered within the area of Lie groups and Lie algebras. – Dietrich Burde Jul 6 '18 at 15:59
• @DietrichBurde if you are referring to the answer I linked, I'm not saying it doesn't. Rather, I'm asking whether that answer can be translated into the language of matrices for someone not familiar with Lie theory – glS Jul 6 '18 at 16:02
• No, you need "solvable" and "simply connected" and other properties from Lie groups, which should not be explained only with matrices, I think. There is topology and algebra, after all, not only matrices. – Dietrich Burde Jul 6 '18 at 16:08
• – user1551 Jul 7 '18 at 4:30