Let $I_{i j}(P)$ be the moment-of-inertia tensor of a rigid body relative to the point $P$. If $G$ is the centre of mass of the body and the vector $G P$ has components $X_{i}$, show that

$$I_{i j}(P)=I_{i j}(G)+M\left(X_{k} X_{k} \delta_{i j}-X_{i} X_{j}\right),$$

where $M$ is the mass of the body.

Consider a cube of uniform density and side $2 a$, with centre at the origin. Find the inertia tensor about the centre of mass, and thence about the corner $P=(a, a, a)$.

Find the eigenvectors and eigenvalues of $I_{i j}(P)$.