We can compute the total angular momentum of a body relative to the point o as follows:

Substituting in the angular velocity term from two slides ago gives:

Now the only term that we have that varies with time is the angular velocity vector. We can factor it out of the summation and the resulting summation is fixed for a given o. We call this term the Inertial tensor (it's just a 3 by 3 matrix, however).

For rigid bodies we will find it convenient to specify the Inertial tensor relative to the object's center of mass.