As derived in the previous section, the moment of inertia tensor, in 3D Cartesian coordinates, is a three-by-three matrix that can be multiplied by any angular-velocity vector to produce the corresponding angular momentum vector for either a point mass or a rigid mass distribution. Note that the origin of the angular-velocity vector is always fixed at in the space (typically located at the center of mass). Therefore, the moment of inertia tensor is defined relative to that origin.
The moment of inertia tensor can similarly be used to compute the mass moment of inertia for any normalized angular velocity vector as
Since rotational energy is defined as (see Eq.(B.7)), multiplying Eq.(B.22) by gives the following expression for the rotational kinetic energy in terms of the moment of inertia tensor:
We can show Eq.(B.22) starting from Eq.(B.14). For a point-mass located at , we have
where again denotes the three-by-three identity matrix, and
which agrees with Eq.(B.20). Thus we have derived the moment of inertia in terms of the moment of inertia tensor and the normalized angular velocity for a point-mass at .
For a collection of masses located at , we simply sum over their masses to add up the moments of inertia:
Now let the mass be located at so that
For , the result is
Principal Axes of Rotation
Angular Momentum Vector