Metric Mapping

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.