Investigation Into Higher Dimensional Rotations



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Axis-angle representations provides efficient methods to study three dimensional rotations. The representation imparts visualization and thus aids the analysis of a three-dimensional proper rotation by reducing its study to that of a two dimensional one. In this dissertation, we accomplish a similar result for five dimensional proper rotation by reducing its study to that of either two or four dimensional proper rotations. For a matrix in SO(5, R), we complete a closed from formula for the axis which is the fixed point set of the matrix as well as the formula for the angle which is the complementary proper rotation that the matrix performs in the orthogonal complement to the axis. In fact, two such derivations are provided. The first is based on the properties of a matrix in SO(5, R) such as the special structure of its characteristic polynomial being skew palindromic while the second utilizes the structure of the Lie algebra of the covering group. Closed form formula for the logarithm in the covering group of SO(5, R) is also derived as it is essential for the second method. Further, we study indefinite rotations with signature (1,9) and come close to establish that the group of such rotations is isomorphic to 2x2 octonion matrices with determinant 1.