数学与应用数学毕业论文——正交矩阵及其应用.doc

本科生毕业设计（论文） 正交矩阵及其应用 学 院： 专 业： 数学与应用数学 学 号： 学生姓名： 指导教师： 二〇一一 年 六 月 摘 要 如果n阶实矩阵A满足,那么称A为正交矩阵．正交矩阵是由内积引出的． 本文例举了正交矩阵在线性代数、化学和物理中的三个应用．在线性代数中,求标准正交基一般用Schimidt正交化方法．本文论证了一种特殊的正交矩阵——初等旋转矩阵——也可以求任一向量空间的标准正交基,并通过实例说明此方法的应用．在化学上,原子轨道的杂化,实际是由一组相互正交的单位基向量,通过线性变换转化为另一组相互正交的单位基向量．而线性代数中由一组标准正交基到另一组标准正交基的过渡矩阵是正交矩阵,因此可以利用正交矩阵的性质求原子轨道的杂化轨道式．在物理上,任一刚体运动都对应一个正交矩阵,本文证明了曲线作刚体运动时曲率和挠率是两个不变量． 关键词：正交矩阵;初等旋转矩阵;标准正交基;原子轨道的杂化;曲率;挠率 Abstract Orthogonal matrices and its applications If a -dimensional real matrix satisfies ,we call it orthogonal matrix. Orthogonal matrix is extracted by inner product. This paper enumerats the applications of orthogonal matrix in linear algebra, chemistry, and physics. Schimidt method is always used to find the standard orthogonal basis in linear algebra. A special kind of orthogonal matrix, namely elementary rotational matrix, is established to find the standard orthogonal basis in this paper. The orbital atom heterozygous is actually made by a team of mutually orthogonal unit basis vector, through linear transformation into another group of mutually orthogonal unit basis in linear algebra. The transition matrix of a group of standard orgthogonal basis to another group of standard orthogonal basis is an orthogonal matrix. Therefore, properties of orthogonal basis can be used to find the orbital atom heterozygous. In physics, any rigid motion corresponds with an orthogonal matrix. The curvature and torsion rate are proved to be two invariants when a curve is in rigid motion. Keywords:Orthogonal matrix; Elementary rotation matrix; Standard orthogonal basis; The orbital atom heterozygous; Curvature;Torsion rate 目 录 1.引 言 1 2.正交矩阵的基本知识 3 2.1正交矩阵的定义与判定 3 2.2 正交矩阵的性质 3 3.正