本文旨在通过解析法严格证明刚体定点运动的等效转轴与瞬时转轴的唯一性，并探究定点运动的欧拉角位移矢量性成立的条件.利用过渡矩阵及其特征向量的性质对等效转轴的唯一性进行了证明，并在此基础上印证了有限欧拉角位移不是矢量.随后证明了瞬时转轴唯一性，并基于过渡矩阵的微分运算得出无限小欧拉角位移为矢量的结论，给出了瞬时转轴方向向量与欧拉角之间的解析关系.基于矩阵运算对定点运动相关结论的严格证明和分析充实和完善了刚体定点运动的描述，证明和分析过程进一步展示了矩阵及其特征值性质在复杂刚体运动分析中的优势.

The purpose of this paper is to rigorously prove the uniqueness of the equivalent and instantaneous rotation axes of the fixed-point motion of a rigid body by the analytical method，and to investigate the conditions under which the Euler angular displacement vectoriality of the fixed-point motion holds. The uniqueness of the e- quivalent axis is proved by using the properties of the transition matrix and its eigenvector，and on this basis，it is proved that the finite Euler angular displacement is not a vector. Then the uniqueness of the instantaneous axis is proved，and based on the differential operation of the transition matrix，it is concluded that the infinitesimal Euler angular displacement is a vector，and the analytic relationship between the direction vector of the instantaneous axis and the Euler angle is given. The rigorous proof and analysis of the conclusions related to the fixed-point motion based on matrix operations enrich and improve the description of the fixed-point motion of a rigid body，and the proof and analysis process further demonstrate the advantages of matrices and their eigenvalue properties in the anal- ysis of complex rigid body motion.