CREATESPACE Quaternions, Biquaternions, and Orthogonal and Unitary Groups (Physics of Spacetime - Complete Collection)

Quaternions were first described by Sir William Rowan Hamilton in the beginning of 19th century. Since then, not a few models in Physics and Engineering were set out based on quaternions.Nowadays, there is a renewed interest in quaternions. On the one hand, by reclaiming the use that James Clerk Maxwell made of them to set out his theory of electromagnetism, more precisely to write his famous equations (the Maxwell equations). And on the other hand, quaternions are efficiently applied in fields like computer graphics, so ubiquitous in audiovisual arts, like cinema, video-games, etc., but also in engineering (where rotations in three-dimensional space are described by means of quaternions, thus providing a key solution to intrinsic problems of other descriptions, like the problem of “gimbal lock”, or loss of a degree of freedom, a very real and serious problem as well as dangerous, in the field of aeronautics and space exploration).This book describes and studies the ring of quaternions and biquaternions, showing how rings of 2×2 matrices as well as 4×4 matrices can be constructed from them. These matrices are called associated matrices. Thus, we can establish an isomorphism between the ring of every type of associated matrix and the ring of quaternions or biquaternions.This book also shows the natural relation between the associated matrices and the groups of orthogonal and unitary matrices (depending on the case), including the special groups SO(2), SO(3), and SU(2). To this end, a detailed introduction of the properties of the groups O(n), SO(n), U(n), and SU(n) is provided.The relation between the associated matrices and other matrices, like the Pauli matrices, is also shown, thus making clear the underlying relation between the latter and the quaternions.