ABSTRACT

In this article, we start by a review of the circle group $\mathbb{T}$ [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle $\left({\mathbb{S}}^1,{\tau }_{{\mathbb{S}}^1}\right)$ . Using points on ${\mathbb{S}}^1$ 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 $G_P\left(\left[\theta \right]\right)$ of projection matrices. Together with the induced topology, it will be demonstrated that $G_P\left(\left[\theta \right]\right)$ is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on $G_P\left(\left[\theta \right]\right)$ to generate subgroups of $G_p\left(\left[\theta \right]\right)$ .

Keywords:

Projection, Orthogonal Projections, Projective Operators, Projective Manifolds

1. Introduction

Orthogonal Projection is a very familiar topic in Linear Algebra [2] . With reference to [2] , it is already known that if V is a finite-dimensional vector space and P is a projection on $W\subset V$ , where W is a subspace of V. Then P is idempotent, that is $P^2=P$ . P is the identity operator on W, that is $\forall x\in W:Px=x$ . We also know that W is the range of P and if U is the kernel of P then $V=W\oplus U,w\in W,u\in U$ . It is easy to show that $w=Px,u=\left(I-P\right)x$ . It is also a known fact that these operators are bounded i.e. $\Vert Px\Vert \le \Vert x\Vert$ . In this paper we will focus on projections in ${ℝ}^2$ and define a different construct for these operators. Starting in the next section with the circle group $\mathbb{T}$ [1] it is possible to endow the set of projective operators with a group and topological structure.

2. Notation Used in This Article

1) $\left({\mathbb{S}}^1,{\tau }_{{\mathbb{S}}^1}\right)$ The unit circle as a Topological Group.

2) $\mathbb{T}$ The circle group defined as $\mathbb{T}:=ℝ/2\pi \mathbb{Z}$ .

3) $\left[\theta \right]$ is an element in the topology of $\mathbb{T}$ .

4) ${\tau }_{ℝ}$ the usual topology on $ℝ$ .

5) ${\tau }_{{\mathbb{S}}^1}$ the topology on the unit circle ${\mathbb{S}}^1$ .

6) $d_{\mathbb{T}}\left(\left[{\theta }_1\right],\left[{\theta }_2\right]\right)$ is the Quotient Metric on $\mathbb{T}$ .

7) $\mathcal{B}\left(\left[\theta \right],ϵ\right)$ opens balls in $\mathbb{T}$ .

8) $\mathcal{B}\left(\theta ,ϵ\right)$ is open in $ℝ$ .

9) $\left(\mathbb{T},\cdot ,\tau \text{\_}\mathbb{T}\right)$ the circle group as a topological group.

10) $\left({\mathbb{S}}^1,\cdot ,{\tau }_{{\mathbb{S}}^1}\right)$ the unit circle as topological group.

11) ${\mathcal{O}}_{{\mathbb{S}}^1}^{{\alpha }_{+}}$ open set in ${\mathbb{S}}^1$ generated by counter-clockwise rotation.

12) ${\mathcal{O}}_{{\mathbb{S}}^1}^{{\alpha }_{-}}$ open set in ${\mathbb{S}}^1$ generated by clockwise rotation.

13) $P_{\left[\theta \right]}$ A projection matrix at angle $\left[\theta \right]$ .

14) $G_p\left(\left[\theta \right]\right)$ the topological projection manifold.

15) ${\mathcal{P}}_{\alpha }$ open sets in $G_p\left(\left[\theta \right]\right)$ .

3. A Brief Review of the Circle Group

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