方向余弦矩阵的特征值与特征向量及其性质研究  

Research on the eigenvalues and eigenvectors of direction cosine matrices and their characteristics

作　　者：张建书[1,2] 陈菲菲 Zhang Jianshu;Chen Feifei(National Key Laboratory of Complex Multibody System Dynamics,Nanjing University of Science and Technology,Nanjing 210094,China;School of Energy and Power Engineering,Nanjing University of Science and Technology,Nanjing 210094,China)

机构地区：[1]南京理工大学复杂多体系统动力学全国重点实验室,江苏南京210094 [2]南京理工大学能源与动力工程学院,江苏南京210094

出　　处：《南京理工大学学报》2025年第1期25-31,共7页Journal of Nanjing University of Science and Technology

基　　金：中央高校基本科研业务费专项资金(30924010302);江苏省自然科学基金(BK20241430)。

摘　　要：用来描述两右手笛卡尔直角坐标系相对方位的方向余弦矩阵是计算力学、计算图形学、导航与控制、机器人等领域中最基本的运算量之一。现有文献已经阐明:2个方位不同的坐标系之间的方向余弦矩阵有3个模为1的特征值,其中一个是实数1,其余2个为一对共轭复数;实特征值1表明两坐标系的相对方位可以通过一次简单转动来描述,与实特征值1对应的特征向量与该简单转动的转轴共线;复特征值指数的模等于该简单转动的转角。该文给出了上述实特征值1对应的特征向量性质的几何证明,并重点研究了方向余弦矩阵复特征值对应的复特征向量的性质,通过证明得出:复特征向量的虚部与实部相互垂直且模相等,虚部与实部的叉积与简单转动的转轴共线且同向。最后通过数值算例对方向余弦矩阵的特征值与特征向量的上述性质进行了验证。The direction cosine matrix,which is used to describe the relative orientation of two right-hand Cartesian rectangular coordinate systems,is one of the most basic operands in computational mechanics,computational graphics,navigation and control,as well as robotics.It has been stated in the existing literature that the direction cosine matrix between two coordinate systems with different orientation has three eigenvalues of unity magnitude,one of which is a real number one,and the other two are a pair of conjugate complex numbers.The real eigenvalue one indicates that the relative orientation of two coordinate systems can be described by a simple rotation,and the corresponding eigenvector is collinear with the rotation axis of the simple rotation.The magnitudes of the exponentials of the complex eigenvalues are equal to the angle of the simple rotation.The property of the eigenvector corresponding to the real number one is proved in a geometrical way in this paper,followed by the main concern of the properties of the complex eigenvectors corresponding to the complex eigenvalues of the direction cosine matrix.It is proved that the imaginary and real parts of the complex eigenvectors of the direction cosine matrix are perpendicular to each other and their magnitudes are equal.The cross product of the imaginary and real parts is collinear to and in the same direction as the axis of simple rotation.Finally,the above properties of eigenvalues and eigenvectors of direction cosine matrix are verified by numerical examples.

关 键 词：方向余弦矩阵 特征值 特征向量 复数特征向量 简单转动 

分 类 号：O302[理学—力学]