REFLECTION DECOMPOSITION OF ROTATION MATRICES
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abstract
Orthogonal matrices are important mathematical objects in astrodynamics mainly because they represent the transformation matrices among reference frames arbitrarily oriented. Examples include spacecraft and planets/moons attitudes as well as the relative orientation of observing instruments with respect to other frames. The fact that the product of two reflections gives an orthogonal matrix is a well known property. Conversely, how to decompose an orthogonal matrix into the product of two reflections is not known. This decomposition is not unique, and this paper provides the whole set of reflection products providing the same orthogonal matrix. This is initially done for the most important 3-dimensional space and it is then extended to any dimensional space. This paper also shows that this decomposition becomes evident in Geometric Algebra, which is briefly summarized. The purpose of this paper is to introduce these mathematical properties and to leave the applications to specific problems such as, attitude estimation, kinematics, decomposition of symmetric and skew-symmetric matrices, to future papers.
