【作者】 武传东；

【导师】 伍俊良；

【作者基本信息】 重庆大学 ， 应用数学， 2009， 硕士

【摘要】 四元数是继复数后又一新的数系,四元数体上代数是复数域上代数的扩展。然而,由于四元数乘法的不可交换性,造成了它与复数域上的代数理论既有一定的联系,又有很大的差别,形成相对独立的内容体系。近年来,四元数代数问题已经引起了数学和物理研究工作者的广泛兴趣。四元数体上代数问题的许多问题已经被研究,比如四元数体上的多项式、行列式、特征值和四元数代数方程组等。然而,四元数体上许多代数问题还需要人们进行进一步的研究,比如四元数矩阵特征值的估计与对角化、四元数矩阵的广义特征值分布与估计、四元数右线性方程组解的扰动性估计问题、四元数矩阵的次亚正定性问题、四元数矩阵方程的可解性问题等等。本文较为系统地分析了四元数体上一些重要的代数特征,主要内容和创新点包括:1.在四元数体上根据特征值的基本概念,将复数域上著名的Gerschgorin圆盘定理推广到四元数体上。由于四元数乘法的不可交换性,得到两种形式的四元数矩阵特征值分布定理,研究了四元数体上严格对角占优矩阵特征值的一些性质。同时给出了四元数矩阵广义特征值的定义,讨论了四元数矩阵左右广义特征值的性质,得到四元数正则矩阵束的广义特征值为实数的结论。获得了估计四元数矩阵广义特征值的Gerschgorin型定理,利用广义瑞利商这一有效的工具,获得了四元数矩阵广义特征值的上下界估计定理。2.对四元数矩阵的对角化进行研究,获得了四元数矩阵可对角化的充要条件,并指出了四元数矩阵的对角化与实(复)数域上矩阵对角化的区别,说明了四元数体上的矩阵性质与实(复)数域上矩阵性质的差异。3.本文在谱半径概念的基础上,讨论了谱半径的估计。4.借助于四元数向量和四元数矩阵的范数理论解决了四元数矩阵求逆、线性方程组的误差估计问题。5.对于次对角线方向上的情形,即四元数体上次亚正定矩阵,本文也作了一些研究,得到一些重要结果。6.矩阵的Kronecker积是一种重要的矩阵乘积,由于四元数乘法的不可交换性,因此,四元数矩阵的Kronecker积性质有所不同于实(复)数矩阵的Kronecker积性质。利用四元数矩阵的Kronecker积这一有效的工具,研究了Lyapunov四元数矩阵方程与Stein四元数矩阵方程的可解性问题。更多还原

【Abstract】 Quaternion is another new number system after the complex. The quaternion field is an extension of complex field. It is a relative independent system and has great difference with the complex field because of the non- commutative multiplication of quaternions.In recent years, the algebra problems over quaternion division algebra have drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalues, and system of quaternion matrices equations and so on.However, there are many questions to be studied in quaternion algebra, such as the estimation of quaternion matrices eigenvalues, diagonalization, the distribution for the generalized eigenvalues of quaternion matrices, the perturbation of solutions of quaternion right linear equations, the sub- positive of quaternion matrices and the solutions of quaternion matrix equations, e.t.c.The purpose of this paper is to discuss some important algebra properties on quaternion division algebra. The main results and innovations are listed as the following:1. The famous Gerschgorin’s disk theorem over complex field is generalized to real quaternion field by the definition of eigenvalues. Two different forms of distribution of the left eigenvalues and right eigenvalues are obtained and some properties of eigenvalues of quaternion diagonally dominant matrices are also discussed. Moreover, the definition of generalized eigenvalues and its properties are given, and then we get a conclusion that the generalized eigenvalue of regular matrix is real. The estimation of the upper and lower bound of quaternion matrices’generalized eigenvalues is obtained with the guidance of the generalized Rayleigh quotient.2. The necessary and sufficient condition for the diagonalized quaternion matrices is given, which is different from real (complex) field. Meanwhile, the difference between diagonalization of complex matrices and quaternion matrices is shown.3. The estimation of spectral radius is discussed based on its concept.4. We solve the quaternion matrices inversion, the estimation of linear equations error and eigenvalues according to quaternion vector and matrices norm theory.5. We also study on the metapositive definite matrices over quaternion division algebra and some results are obtained.6. The Kronecker product is important. The quaternion field is different from the complex field because of the non- commutative multiplication of quaternions. Then we study the solutions of quaternion Lyapunov matrix equations and the Stein Equations by Kronecker product over quaternion division algebra.更多还原