I understand that the inertia tensor is basically a 3x3 matrix. I'm modelling a cube, and the inertia tensor for this shape has values for moment of inertia on the main diagonal of the matrix. All the products of inertia are zero.
However, I understand that this matrix only makes sense in body (i.e., local) coordinates. That is if I have a cube and rotate it, its local coordinate axes rotate with it and the inertia tensor remains constant. However, I need to represent the inertia tensor in world coordinates as well. How can I do that? Do I need to multiply it by the current orientation? I'm using quaternions for orientation, so I can extract a matrix from the quaternion (a 4x4 matrix, and if I get rid of the fact that a 4x4 matrix represents homogeneous coordinates I could convert it into a 3x3. To do this I would simply get rid of the rightmost column and bottommost row of the 4x4 matrix returned from the quaternion).
I hope this is making some sense. I suspect that the values of the inertia tensor in world coordinates will change as the orientation of the cube changes.
Is my logic correct? Formulas would be helpful.