## Matrix Groups are Manifolds

We spent a lot of time talking about manifolds in the Lie Group workbook. Manifolds are an invaluable tool for studying anatomical shapes because
they represent very important *intrinsic* properties of the shape.

We can show that every Matrix Group has a corresponding Lie algebra, defined as the tangent space around the identity element of the matrix group. Because we know that Lie groups are manifolds, it seems that there would be a way to convert matrix groups, with corresonding Lie algebras, to manifolds.

The way to do this is with a parameterization of the matrix group that for every point in the group maps an open set of the real space into a local neighborhood of the matrix. As it turns out, the operation of matrix exponentiation provides just such a mechanism.