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 ALAMT  Vol.9 No.1 , March 2019
A Follow-Up on Projection Theory: Theorems and Group Action
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.
Cite this paper: Niglio, J. (2019) A Follow-Up on Projection Theory: Theorems and Group Action. Advances in Linear Algebra & Matrix Theory, 9, 1-19. doi: 10.4236/alamt.2019.91001.
References

[1]   Tapp, K. Matrix Groups for Undergraduates. ISBN 0-8218-3750-0.
http://www.ams.org/publications/authors/books/postpub/stml-29

[2]   Yanai, H., Takeuchi, K. and Takane, Y. Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition. ISBN 1441998861, Springer.

[3]   Rudolph, G. and Schmidt, M. (2103) Differential Geometry and Mathematical Physics: Part I Manifold, Lie Groups and Hamiltonian Systems. ISBN 978-94-007-5344-0, Springer, Berlin.

[4]   Levine, M. GLn(R) as a Lie Group. University of Chicago.
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Levine.pdf

 
 
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