If $\{A_i\}$ is a sequence in the endomorphism of $\mathbb R^n$ such that $e^{A_i}$ converges to $e^A$, then can we conclude that $A_i$ converges to $A$?
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2Hint: In this question you will find many examples of matrices $A$ such that $e^A = I$. – Djaian Apr 05 '13 at 06:59
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No. For $n>1$, the matrix exponential function is not injective on $\mathbb{R}^{n\times n}$. Therefore there exist $A\neq B$ such that $e^A=e^B$. Take $A_i=B$ for all $i$, you get a counterexample. For instance, take $A=0$ and $B=\pmatrix{0&-2\pi\\ 2\pi&0}$. Then $e^A=e^B=I$. For $n=2$, see a related question here.
