1. State the conditions under which a point and the mass center of a rigid body share the same principal axes of inertia. In other words, let Ip and le be the moment-of-inertia matrices of a rigid body about a point P and its mass center, C, respectively. State the conditions under which the two matrices have common eigenvectors. Moreover, under these conditions, what are the relationships between the two sets of principal moments of inertia?
2. Show that the smallest principal moment of inertia of a rigid body attains its minimum value at the mass center.