Matrix Lie group

A matrix Lie group is a subgroup of a general linear group that is also a Lie group for its matrix topology. Near the identity, the matrix exponential and matrix logarithm give inverse coordinate charts whenever they are restricted to sufficiently small neighborhoods in the group and its tangent space.

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.

For a matrix Lie group $G$ with Lie algebra $\mathfrak{g}=T_{I}G$, choose neighborhoods on which $\exp :\mathfrak{g}\to G$ and $\log :G\to \mathfrak{g}$ are inverse. Left translation gives a chart near $g\in G$ byThe Baker--Campbell--Hausdorff formula expresses multiplication smoothly in these coordinates.