四元数的表示以及四元数矩阵的秩
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摘要
虽然从1843年Hamilton发现四元数至今已经过去不止一个半世纪了，但四元数代数的某些方面仍然是神秘的，有待人们的探索。因此，对四元数的研究到现在仍然是数学中的一个活跃领域也就不足为奇了。在本论文中，我们研究了四元数的实表示，复表示的某些方面，以及四元数矩阵的秩。全文由五章组成，包括了如下内容。
第一章综述了四元数理论的发展以及必要的基本知识。
在第二章中，受Farebrother等人工作的启示，我们研究了四元数的实矩阵表示，给出了使得虚单位对应的矩阵是带符号置换矩阵的所有可能的实矩阵表示。我们考虑了四元数代数的生成元（即虚单位）的象，分析了其性质，然后确定了什么样的实矩阵满足这些性质。运用群作用的语言，我们得到了有趣的结论：四元数代数的一般表示的矩阵对子在本质上仅由两个基本的4乘4实矩阵对子组成。
在第三章中，我们给出了捷联惯性导航系统的四元数姿态微分方程的一个复表示。接下来，我们引入了航天器捷联惯性导航系统的复维数为2的姿态算法，从而减小了解算四元数姿态微分方程的计算复杂度。我们的这种新方法不但有助于改进解算四元数姿态微分方程的效率，而且对进行实时计算也有所助益。
在第四章中，我们研究了四元数矩阵的左、右秩。对于一个四元数矩阵，该矩阵本身与其转置矩阵，共轭矩阵的左、右秩之间的关系首先得到澄清。我们指出，四个自然的关于左、右秩的问题都等价于如下问题：对什么样的矩阵，其左、右列秩相等？为了回答这个问题，我们首先把它归结为寻找某些四元数矩阵方程的解集。通过分析体上可逆矩阵的结构，我们给出了当所涉及的矩阵是简单矩阵时的一些有趣的例子。最终，通过把这些矩阵方程转化为某种特殊形式，我们获得了满足上述条件的矩阵的显式表达，从而使本章的主要问题得到了完满解决。
在第五章中，基于对一般域和一般体上有限维向量空间之间差别的观察，我们引入了四元数向量的一个新的特征数，即高度。根据这个概念，四元数向量被分成五个类型，其中每个类型有特定的高度。本着考察高度，我们又给出了四元数向量的关联矩阵的概念。而后，我们注意到一个四元数向量的关联矩阵可能的行阶梯形比一般的四元数矩阵本质上要少。在这方面，我们给出了四元数向量的一个完整的分类，并列出了每种类型的向量的关联矩阵所有可能的行阶梯形。对于某些类型的四元数向量，我们各自举出了非平凡的例子。最后，通过观察高度的概念与笛卡尔标架序列之间的联系，我们给出了对若干类型的笛卡尔标架序列的特征描述。
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.
第一章综述了四元数理论的发展以及必要的基本知识。
在第二章中，受Farebrother等人工作的启示，我们研究了四元数的实矩阵表示，给出了使得虚单位对应的矩阵是带符号置换矩阵的所有可能的实矩阵表示。我们考虑了四元数代数的生成元（即虚单位）的象，分析了其性质，然后确定了什么样的实矩阵满足这些性质。运用群作用的语言，我们得到了有趣的结论：四元数代数的一般表示的矩阵对子在本质上仅由两个基本的4乘4实矩阵对子组成。
在第三章中，我们给出了捷联惯性导航系统的四元数姿态微分方程的一个复表示。接下来，我们引入了航天器捷联惯性导航系统的复维数为2的姿态算法，从而减小了解算四元数姿态微分方程的计算复杂度。我们的这种新方法不但有助于改进解算四元数姿态微分方程的效率，而且对进行实时计算也有所助益。
在第四章中，我们研究了四元数矩阵的左、右秩。对于一个四元数矩阵，该矩阵本身与其转置矩阵，共轭矩阵的左、右秩之间的关系首先得到澄清。我们指出，四个自然的关于左、右秩的问题都等价于如下问题：对什么样的矩阵，其左、右列秩相等？为了回答这个问题，我们首先把它归结为寻找某些四元数矩阵方程的解集。通过分析体上可逆矩阵的结构，我们给出了当所涉及的矩阵是简单矩阵时的一些有趣的例子。最终，通过把这些矩阵方程转化为某种特殊形式，我们获得了满足上述条件的矩阵的显式表达，从而使本章的主要问题得到了完满解决。
在第五章中，基于对一般域和一般体上有限维向量空间之间差别的观察，我们引入了四元数向量的一个新的特征数，即高度。根据这个概念，四元数向量被分成五个类型，其中每个类型有特定的高度。本着考察高度，我们又给出了四元数向量的关联矩阵的概念。而后，我们注意到一个四元数向量的关联矩阵可能的行阶梯形比一般的四元数矩阵本质上要少。在这方面，我们给出了四元数向量的一个完整的分类，并列出了每种类型的向量的关联矩阵所有可能的行阶梯形。对于某些类型的四元数向量，我们各自举出了非平凡的例子。最后，通过观察高度的概念与笛卡尔标架序列之间的联系，我们给出了对若干类型的笛卡尔标架序列的特征描述。
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.
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