Geometrical transformations in higher dimensional Euclidean spaces

Abstract

Orientations and rotations in n-dimensional real Euclidean spaces (R[n]) are represented by proper orthogonal, or skew-symmetric matrices. A mathematical formulation that leads to these representations is presented. Orientations and rotations are indistinguishable in 2 and 3 dimensions. In higher dimensions, orientations can be achieved by a minimal set of rotations. This result is presented here as the generalization of Euler's Principal Rotation Theorem to higher dimensions. Three types of skew-symmetric orientation and rotation matrices are presented. Decompositions of orientation matrices, in terms of rotation matrices, are also presented. Comparisons are drawn between these matrix representations of rotations and orientations. The ortho-skew matrices, which are both orthogonal and skew-symmetric, are introduced as a special set of orientation matrices. Symmetric matrices often arise in linear systems theory and estimation. They represent reflections and projections (both orthogonal and non-orthogonal), in Euclidean spaces. The ortho-symmetric matrices, which are both orthogonal and symmetric, are introduced. These matrices represent reflections in Euclidean spaces. The Householder matrices, often encountered in linear algebra problems, belong to this set and represent elementary reflections. A general symmetric matrix can be decomposed as a sum of scalar multiples of a set of Householder matrices. Elementary projections in R[n] can be represented by a set of symmetric matrices, called the modified Householder matrices, introduced here. These matrices are a natural choice for decomposing symmetric matrices. This decomposition closely parallels the decomposition of orientation matrices by rotation matrices. The last part of this thesis deals with the matrix Riccati differential equation with symmetric coefficients, also known as the symplectic matrix Riccati differential equation (SRDE). This equation, along with the related but simpler Lyapunov equation, arises quite frequently in optimal control theory and estimation theory. A solution procedure, which solves the time-varying SRDE by extension to a symplectic flow field, and utilizes the properties of symplectic matrices, is presented here. This solution can be related to the analytic singular value decomposition of the time-varying symmetric matrix solution.