【作者】 凌思涛；

【导师】 姜同松；

【作者基本信息】 曲阜师范大学 ， 运筹学与控制论， 2007， 硕士

【摘要】 几类四元数矩阵方程的解及其算法研究四元数和四元数矩阵不但在稳定性理论和控制论等诸多方面的应用十分广泛，它们也是研究量子力学的重要工具，四元数矩阵方程则是研究四元数量子力学理论中相应的数学物理模型的基础。由于四元数乘法的非交换性，使得一些四元数矩阵方程很难求解，因此，研究非交换意义下四元数量子力学的数学物理模型解是一项十分有意义的工作，对促进现代力学的发展会起到很大的作用。本文主要对于线性和非线性四元数矩阵方程解的算法作了一些研究，给出了几类四元数矩阵方程解的表达式。作为第三章和第四章的理论基础，我们在第二章中介绍了关于四元数矩阵的一系列的代数性质，然后给出四元数矩阵可以次对角化的充要条件及其相应的算例。在第三章中，我们利用四元数矩阵的广义奇异值分解，研究了四元数线性矩阵方程AXA~*=B的(反)中心对称解及最佳逼近解。在第四章中，我们利用四元数矩阵的复表示和四元数矩阵的Jordan标准形理论，结合矩阵方程解的性质给出了一类四元数非线性方程解的算法及其解的一般通式。更多还原

【Abstract】 Quaternions and quaternion matrices have extensive applications in stability theory and cybernetics, they are important tools in studying quantum mechanics, while quaternion matrix equations are the basis of corresponding mathematical physics models in studying quaternionic quantum mechanics. Some quaternion matrix equations are difficult to solve because of noncommutation of quaternion multiplication, so it is a quite meaningful thing that study solutions of mathematical physics models in quaternionic quantum mechanics under noncommutation.In this dissertation, we mainly do some studies in algorithms for linear and nonlinear quaternion matrix equations, and present expressions for solutions of some quaternion matrix equations.As the theoretical basis of Chapter 3 and Chapter 4, a series of algebraic properties of quaternion matrices are introduced, then necessary and sufficient conditions for secondary diagonalization matrices and a corresponding example are given in Chapter 2.In Chaper 3, by means of quaternion generalized singular value decomposition of quaternion matrices, we obtain a general expression that quaternion matrix equation AX A~* = B has centro-symmetric solutions or anti-centro-symmetric solutions, in addition, an expression for the optimal approximation solution to a given matrix is derived.In Chaper 4, using methods of complex representation and theories of Jordan canonical forms of quternion matrices, combining properties of solutions of matrix equations, we give an algorithm for a class of nonlinear quaternion matrix equations and present general expressions for solutions of them.更多还原