四元数矩阵代数中的若干问题研究
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摘要
本论文以四元数矩阵代数为研究背景，分别讨论了四元数矩阵重行列式的估计、多个四元数矩阵的同时对角化、四元数矩阵Jordan分解变换阵的算法以及四元数矩阵方程的求解等问题，所获得的结果推广和改进了已有文献的结论，是四元数矩阵理论和直用的新发展全文分为五章，具体内容介绍如下：
在第一章绪论中简述了四元数及四元数矩阵理论的背景、发展状况以及本文的主要结果
在第二章中，研究了两个四元数矩阵的和的重行列式的估计一方面，通过构造性的方法，给出了任意两个四元数矩阵的和的重行列式的上界估计；另一方面，对两个构成协调对的四元数矩阵，建立了它们的和的重行列式的下界估计式同时注意到，一个数域上的矩阵也是一个四元数矩阵，从而利用本章的结论可建立一系列数域上矩阵行列式的相关不等式最后，作为直用，举例说明了一些现有的结论可作为本章结论的推论，并完全地回答了复矩阵行列式理论中的一个问题
在第三章中，从两个方面研究了多个四元数矩阵的同时对角化问题首先，给出了一个四元数矩阵对同时实对角化的定义，讨论了两个四元数矩阵可同时实对角化的充分必要条件，并给出了可行的算法进一步，讨论了一个四元数长矩阵集可同时实对角化的情况作为直用，我们将四元数矩阵对的同时实对角化直用于求解四元数矩阵方程中，所获得的结果也推广了现有文献的结果其次，给出了四元数矩阵对的同时复对角化的定义，讨论了两个四元数长矩阵的同时复对角化的充分必要条件进一步，对一个四元数长矩阵集，针对其同时复对角化问题，给出了一系列充分必要条件
在第四章中，通过引入四元数矩阵Jordan链的定义，结合四元数矩阵的复导出阵与四元数矩阵之间的对直关系，给出了一个计算任意四元数矩阵的Jordan标准形变换阵的完全算法，它为四元数矩阵代数在更广泛的领域中获得直用提供了基础
在最后一章，由于常规的矩阵Kronecker积的公式对于一般的四元数矩阵不再成立，因此用常规的手段来研究四元数矩阵方程通常是失效的首先，研究了四元数体上一类广义sylvester方程，AX-xB=0，的所有解的具体表达式其次，通过将复矩阵方程看作四元数矩阵方程，从而利用四元数矩阵独有的性质来给出了复矩阵方程Ax-xB=c有解的充分必要条件，进一步给出了求解此方程的具体算法最后，我们讨论了四元数体上一类特征值反问题及其最小二乘问题，推广了现有文献的结果
This dissertation focuses on some problems existing in the quaternion matrix al-gebra and its applications. In this dissertation, we study the estimation of the doubledeterminant of quaternion matrices, simultaneous diagonalization of quaternion matrices,algorithm for computing the transition matrices of Jordan canonical form over quaternionskew-field and solving quaternion matrix equations. The dissertation is divided into fivechapters and main contents are as follows:
In chapter one, the preface, we introduce a survey to the development of quaternionmatrix algebra and the background of quaternion and quaternion matrices. Meanwhile,main results in this dissertation are summarized.
In chapter two, we study the estimation of the double determinant of the sum of twoquaternion matrices. Firstly, an estimation of the upper bound is given for the double de-terminant of the sum of two arbitrary quaternion matrices by constructive way. Secondly,the lower bound on the double determinant is established especially for the sum of twoquaternion matrices which form an assortive pair. Note that a matrix over number field isalso a quaternion matrix, hence a series related inequalities over the number fields can beestablished. Finally, as applications, some known results are obtained as corollaries and aquestion in the matrix determinant theory is answered completely.
Chapter three is divided into two sections. In the first section, the definition of simul-taneous real diagonalization of a pair of quaternion matrices is founded, some necessaryand suficient conditions are discussed for two quaternion matrices can be simultaneouslyreal diagonalized, and an algorithm for computing the simultaneous real diagonalizationof two quaternion matrices is provided. Finally, the simultaneous real diagonalization ofa pair of quaternion matrices is applied in solving quaternion matrix equations. In thesecond section, the definition of simultaneous complex diagonalization of a pair of quater-nion matrices is established, and necessary and suficient conditions of two quaternionmatrices can be simultaneously complex diagonalized are obtained. Moreover, a series ofnecessary and suficient conditions are established for the set of quaternion rectangularmatrices which can be simultaneously complex diagonalized.
In chapter four, by the definition of Jordan chains over quaternion skew-field, and bythe corresponding relationship between quaternion matrices and their complex derived ma-trices, a complete algorithm for computing the transition matrices of the Jordan canonical forms of quaternion matrices is given.
Note that, since the formulae of Kronecker product over number fields hold no longerfor quaternion matrices, it is often not convenient to handle the quaternion matrix equa-tions cannot be studied by the usual way. In the last chapter, firstly, we study one kindof generalized Sylvester equations, AX fi XB = 0 , and give out the concrete expressionof the general solutions. Secondly, since complex matrix equations are also quaternionmatrix equations, we study the complex matrix equation AX - XB = C . By the spe-cial properties of quaternion matrices, necessary and suficient conditions are obtained forthe matrix equation being consistent. Furthermore, a complete algorithm for solving thisequation is given. Finally, we study one kind of inverse problems and its least-squaresproblems, which improve the existing results also.
在第一章绪论中简述了四元数及四元数矩阵理论的背景、发展状况以及本文的主要结果
在第二章中，研究了两个四元数矩阵的和的重行列式的估计一方面，通过构造性的方法，给出了任意两个四元数矩阵的和的重行列式的上界估计；另一方面，对两个构成协调对的四元数矩阵，建立了它们的和的重行列式的下界估计式同时注意到，一个数域上的矩阵也是一个四元数矩阵，从而利用本章的结论可建立一系列数域上矩阵行列式的相关不等式最后，作为直用，举例说明了一些现有的结论可作为本章结论的推论，并完全地回答了复矩阵行列式理论中的一个问题
在第三章中，从两个方面研究了多个四元数矩阵的同时对角化问题首先，给出了一个四元数矩阵对同时实对角化的定义，讨论了两个四元数矩阵可同时实对角化的充分必要条件，并给出了可行的算法进一步，讨论了一个四元数长矩阵集可同时实对角化的情况作为直用，我们将四元数矩阵对的同时实对角化直用于求解四元数矩阵方程中，所获得的结果也推广了现有文献的结果其次，给出了四元数矩阵对的同时复对角化的定义，讨论了两个四元数长矩阵的同时复对角化的充分必要条件进一步，对一个四元数长矩阵集，针对其同时复对角化问题，给出了一系列充分必要条件
在第四章中，通过引入四元数矩阵Jordan链的定义，结合四元数矩阵的复导出阵与四元数矩阵之间的对直关系，给出了一个计算任意四元数矩阵的Jordan标准形变换阵的完全算法，它为四元数矩阵代数在更广泛的领域中获得直用提供了基础
在最后一章，由于常规的矩阵Kronecker积的公式对于一般的四元数矩阵不再成立，因此用常规的手段来研究四元数矩阵方程通常是失效的首先，研究了四元数体上一类广义sylvester方程，AX-xB=0，的所有解的具体表达式其次，通过将复矩阵方程看作四元数矩阵方程，从而利用四元数矩阵独有的性质来给出了复矩阵方程Ax-xB=c有解的充分必要条件，进一步给出了求解此方程的具体算法最后，我们讨论了四元数体上一类特征值反问题及其最小二乘问题，推广了现有文献的结果
This dissertation focuses on some problems existing in the quaternion matrix al-gebra and its applications. In this dissertation, we study the estimation of the doubledeterminant of quaternion matrices, simultaneous diagonalization of quaternion matrices,algorithm for computing the transition matrices of Jordan canonical form over quaternionskew-field and solving quaternion matrix equations. The dissertation is divided into fivechapters and main contents are as follows:
In chapter one, the preface, we introduce a survey to the development of quaternionmatrix algebra and the background of quaternion and quaternion matrices. Meanwhile,main results in this dissertation are summarized.
In chapter two, we study the estimation of the double determinant of the sum of twoquaternion matrices. Firstly, an estimation of the upper bound is given for the double de-terminant of the sum of two arbitrary quaternion matrices by constructive way. Secondly,the lower bound on the double determinant is established especially for the sum of twoquaternion matrices which form an assortive pair. Note that a matrix over number field isalso a quaternion matrix, hence a series related inequalities over the number fields can beestablished. Finally, as applications, some known results are obtained as corollaries and aquestion in the matrix determinant theory is answered completely.
Chapter three is divided into two sections. In the first section, the definition of simul-taneous real diagonalization of a pair of quaternion matrices is founded, some necessaryand suficient conditions are discussed for two quaternion matrices can be simultaneouslyreal diagonalized, and an algorithm for computing the simultaneous real diagonalizationof two quaternion matrices is provided. Finally, the simultaneous real diagonalization ofa pair of quaternion matrices is applied in solving quaternion matrix equations. In thesecond section, the definition of simultaneous complex diagonalization of a pair of quater-nion matrices is established, and necessary and suficient conditions of two quaternionmatrices can be simultaneously complex diagonalized are obtained. Moreover, a series ofnecessary and suficient conditions are established for the set of quaternion rectangularmatrices which can be simultaneously complex diagonalized.
In chapter four, by the definition of Jordan chains over quaternion skew-field, and bythe corresponding relationship between quaternion matrices and their complex derived ma-trices, a complete algorithm for computing the transition matrices of the Jordan canonical forms of quaternion matrices is given.
Note that, since the formulae of Kronecker product over number fields hold no longerfor quaternion matrices, it is often not convenient to handle the quaternion matrix equa-tions cannot be studied by the usual way. In the last chapter, firstly, we study one kindof generalized Sylvester equations, AX fi XB = 0 , and give out the concrete expressionof the general solutions. Secondly, since complex matrix equations are also quaternionmatrix equations, we study the complex matrix equation AX - XB = C . By the spe-cial properties of quaternion matrices, necessary and suficient conditions are obtained forthe matrix equation being consistent. Furthermore, a complete algorithm for solving thisequation is given. Finally, we study one kind of inverse problems and its least-squaresproblems, which improve the existing results also.
引文
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