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.