Considering this post: Show the exponential map $\mathfrak{u}(n)\rightarrow U(n)$ is surjective I was trying to prove that the exponential map $ \mathfrak{su(n)} \longrightarrow SU(n)$ is surjective, where:
$$ \mathfrak{su(n)}:= \{ A \in U(n) \space | \space Tr(A)=0 \space \} $$
$$ SU(n) := \{ A \in U(n) \space | \det(A)=1 \}$$
but I dont know where to start, I thought about restricting the eigenvalues of a matrix, analogous to the $D_X$ in the post but can quite get it. How can we do this?