Representations Of Quaternions And Ranks Of Quaternionic Matrices

Although more than one and a half centuries have passed since the discovery ofquaternions by W. R. Hamilton in1843, some aspects of quaternion algebra remainmysterious and to be explored. Therefore, it is not surprising that the research ofquaternions is an active field in mathematics until recent years. In this dissertation westudy some aspects on real and complex representations of quaternions, including ranksof quaternionic matrices. The dissertation consists of five chapters, including thefollowing contents.Chapter1summarizes the development and the necessary basic knowledges of thetheory of quaternions.In Chapter2, inspired by the work of R. W. Farebrother et al., we investigate the realmatrix representations of quaternions, and find all the possible representations under thecondition that the matrices of the imaginary units are signed permutation matrices. Weconsider the images of the generators (i.e. the imaginary units) of quaternion algebra,analyze the properties of the images, and then determine what kind of real matricesfulfill them. By the language of group action, the conclusion turns out to be interesting:the general representation pairs of matrices for the quaternion algebra is essentiallymade up of only two basic pairs of4×4real matrices.In Chapter3, a complex representation is established for the main quaternionicattitude differential equation in Strap-down Inertial Navigation System (SINS).Consequently, an attitude algorithm of complex dimension2is introduced for SINS ona spacecraft and the computational complexity of solving the main quaternionic attitudedifferential equation is reduced. This new method is helpful not only to improve theefficiency of solving the quaternionic attitude differential equation, but also to performa real-time computation.In Chapter4, we study the left and right ranks of a quaternionic matrix. Therelationships among the left and right ranks of a quaternionic matrix, its transpose, andits conjugate are clarified. We point out that four rank related questions naturally arisedare equivalent to the following one: what matrices have the same left and right columnranks? To answer this question, we reduce it to finding the solution sets of some kind ofquaternionic matrix equations. By analyzing the structure of the invertable matrices overa skew-field, we give some interesting examples when the matrices involved are simplematrices. Finally, by transforming the quaternionic matrix equations to a special form,we obtain an explicit representation for such matrices, and thus the main question of thischapter is answered completely.In Chapter5, based on the observation of the differences between thefinite-dimensional vector spaces over a general field and skew-field, we introduce a new numerical character, namely the height, for a quaternionic vector. According to thisconcept, the quaternionic vectors are divided into five types, each of which has aspecified height. Concerning the height, the associated matrix of a quaternionic vector isdefined. We then notice that the possible row echelon forms of the associated matrix ofa quaternion vector is essentially less than that of a general quaternion matrix. In thisrespect, we give a complete classification of quaternionic vectors, and enumerate allpossible row echelon forms of the associated matrices for each type of quaternionicvectors. Respective non-trivial examples are given for some of the types of quaternionicvectors. Finally, by observing a link which the height has with the sequence of Cartesianframes, we give the character descriptions for several types of the sequences ofCartesian frames.