Abstract: The moment of inertia is the measure of the rotational inertia of the rigid body, and the rotational inertia center is the special rotational center of the rigid body. The inertia tensor to the rotation center is a secondorder spherical tensor. The moment of inertia of a rigid body is equal to any axis passing through the rotational inertia center, and any axis passing through the rotation center is the principal axis of inertia. The theorem for the rotation inertia center is proposed and proved, namely,  when the three principal moments of inertia to the center of mass are equal, the center of mass is the only rotation center; When only two smaller principal moments of inertia are equal, the rigid body has only two rotation centers, both on the principal axis of larger principal moment and symmetrically distributed with respect to the center of mass, and the distance from the two rotation centers to the center of mass is equal to the square root of the ratio of the difference between the larger and smaller principal moment of inertia to the mass of the rigid body; When the three principal moments of inertia are not equal to each other or only two larger of them are equal, there is no rotation inertia center.

Key words: rotation inertia center, moment of inertia, principal axis of inertia, principal moment of inertia, inertia tensor