%0 Journal Article
%T The Projective Group as a Topological Manifold
%A Jean-Francois Niglio
%J Advances in Linear Algebra & Matrix Theory
%P 134-142
%@ 2165-3348
%D 2018
%I Scientific Research Publishing
%R 10.4236/alamt.2018.84012
%X In this article, we start by a review of the circle group&nbsp;<img alt=\"\" src=\"Edit_56b67c89-d649-4b40-a7e5-52dfe08ce857.png\" /> [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle <img alt=\"\" src=\"Edit_ebd1e9fa-b8ae-4938-8b07-190c9ac2f3b3.png\" />. Using points on&nbsp;<img alt=\"\" src=\"Edit_91405653-d8a3-4f20-8a89-ccb244dce4e9.png\" /> under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted&nbsp;<img alt=\"\" src=\"Edit_4546d882-7c35-4bba-859b-457c366c14d1.png\" /> of projection matrices. Together with the induced topology, it will be demonstrated that&nbsp;<img alt=\"\" src=\"Edit_83a852cb-17f4-41a9-b03e-3130787fcdb2.png\" /> is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on&nbsp;<img alt=\"\" src=\"Edit_8c601c48-dbc3-4d2f-b536-48b5db4dc74c.png\" /> to generate subgroups of&nbsp;<img alt=\"\" src=\"Edit_fc8aea47-fcd8-4575-9a62-0682eaf1d87f.png\" />.
%K Projection
%K Orthogonal Projections
%K Projective Operators
%K Projective Manifolds
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=89008