Exponential map of a matrix Lie group ($\exp :\mathfrak{g}\to G$)

For a Matrix Lie group $G$ with Lie algebra $\mathfrak{g}$, its exponential map is the restriction of the matrix exponential, $X\mapsto e^X$. It maps $\mathfrak{g}$ into $G$ and is a local diffeomorphism at zero, but it need not be surjective globally.