Inertia tensor ($I_{ij}$)

The inertia tensor about a point isIt maps angular velocity to angular momentum.

The principal moments of inertia are the eigenvalues of the inertia tensor, measured along its principal axes.

For torque-free rigid-body motion, the kinetic-energy ellipsoid fixed in the body rolls without slipping on the invariable plane perpendicular to the conserved angular momentum. Its point of contact is the angular-velocity vector.