【作者】 徐渤；

【导师】 冯良贵；

【作者基本信息】 国防科学技术大学 ， 数学， 2007， 硕士

【摘要】 本文以实数和复数域中关于矩阵分解和矩阵方程的既有理论为线索,研究四元数体上的矩阵分解和矩阵方程理论,把实数和复数域上部分关于矩阵分解和矩阵方程的结论推广到了四元数体上,得到了一些有意义的结果,深化了人们对四元数矩阵的认识,丰富了四元数矩阵理论。本文的主要成果在于:1.通过定义四元数矩阵的一种新的LU分解,将四元数矩阵的LU分解和复数矩阵的LU分解联系起来,结果获取了该新的LU分解存在性的判定条件和获取该分解的计算方法。2.证明了四元数矩阵有唯一LDU分解的充分必要条件,把对复数矩阵进行Crout分解和Doolittle分解的实用算法推广到四元数矩阵上,并给出了算例。3.把复数矩阵方程AXB=C的一些性质和结论逐一推广到四元数矩阵上。在A和B中一个是实数矩阵另一个是自共轭阵的情况下,得到了四元数矩阵方程AXB=C有解的充分必要条件和解的构造。更多还原

【Abstract】 We studys the theory of matrix decomposition and matrix equation over quaternion field, along the clues of the theory of matrix decomposition and matrix equation over real number field and complex number field. It gets some results and enrichs quaternion matrices theory.The main results of this paper lie in:1. By introducing a new LU decomposition of quaternion matrix, we establish the relationship between this new LU decomposition of quaternion matrix and the corresponding part of complex matrix. Consequently, we obtain this new LU decomposition existence determinant condition and obtain the method of calculation.2. We give the necessary and sufficient condition of quaternion matrices to have unique LDU decompositions and build a kind of practical algorithm of matrix LU decompositions, over including Crout decompositions and Doolittle decompositions.3. We extend some properties and conclusions of complex number matrix equation AXB = C over quaternion field and give the necessary and sufficient condition of which the quaternion matrix equation AXB = C can be solved. Moreover, we describe the structure of solutions of such an equation under circumstances that one of A and B is real matrix, other is self-conjugate matrix.更多还原