Abstract
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on by using the rotation group [3] [4]. It will be proved that the group acts on elements of in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.
Highlights
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory
We have already established that one-parameter projection operators can be constructed from the unit circle 1
( ) fine the group operation φ P[θ], P[θ′] in terms of matrix operations using the Vec operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article
Summary
In this article the notation has been kept the same as in the previous article. ⋅ F is the Frobenius Norm. In this article the notation has been kept the same as in the previous article. P[θ] is some projection operator in GP ([θ ]). ( ) Rnk P[θ] is the Rank of a projection operator. ( ) σ P[θ] is the spectrum of a projection operator. ( ) StabSO(2) P[θ] is the Stabilizer subgroup for some fixed element of GP ([θ ]). R (α ) is an element of SO (2) for angle α.
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