Inertia tensor and principal axes

Summary

The inertia tensor of a rigid body about a point is a symmetric $3 \times 3$ matrix. It collects the moments of inertia of the body about all axes through that point. It turns the angular velocity of the body into its angular momentum, and it gives the kinetic energy of rotation as a quadratic form. Its eigenvectors span the principal axes of the body, and its eigenvalues are the principal moments of inertia. In a frame along the principal axes the equations of rigid body motion take their simplest form.