TITLE: Finding a vector subspace
QUESTION [1 upvotes]: Considering the exponential map $\exp : M_{n*n}(\mathbb{C}) → GL(n, \mathbb{C})$ we know that it is a local diffeomorphism at $0$.
How can we find the appropriate vector subspaces $V( U(n))$ such that the restriction:
$$ \exp: V(U(n)) \longrightarrow U(n)$$
is a local diffeomorphism at $0$?
My attempt:
Choosing $A \in V(U(n))$ and writing it in its eigenbasis it will only have elements, whose norm is $1$, in the main diagonal so its easy to see that:
$$ \exp(-A) = [\exp(A)]^{-1}=[\exp(A)]^{\dagger}=\exp(A^{\dagger})$$
Applying the exponential map to:
$$ A^{\dagger}+A=0 \longrightarrow \exp(A^{\dagger}+A)= \exp(A^{\dagger}) \exp(A) = [\exp(A)]^{\dagger}\exp(A)=\exp(-A)\exp(A) = \exp(A-A)= I$$
then, the restriction that we must apply is:
$$ A^{\dagger}+A=0$$
Is my attempt correct?

REPLY [0 votes]: Hint: For $A\in V(U(n))$, you want
$$ \exp(-A)=\exp(A)^{-1}=\exp(A)^H=\exp(A^H)$$
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