Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 2 / 2C / Solution

Let $M_{ij}={\int }_S{x}_i{x}_{j}dS$. A rotation matrix $Q$ maps the unit sphere and its surface measure to themselves. Changing variables therefore gives $Q_{ip}{Q}_{jq}{M}_{pq}=M_{ij}$, so $M$ is an isotropic second-rank tensor. Reflection symmetry also directly makes its off-diagonal entries zero, while interchange of coordinates makes its three diagonal entries equal. Its trace ishence $M_{ij}=(4\pi /3){\delta }_{ij}$. Oddness gives ${\int }_S{x}_{i}dS=0$. Expanding the defining product for $T$ now givesThe action on a vector is $Tv=\lambda v+\mu y(y\cdot v)$. For $y\ne 0$, its eigenvalues and eigenspaces are thereforeThe latter has multiplicity two. If $y=0$, the single eigenvalue $4\pi /3$ has multiplicity three and every vector is an eigenvector apart from the excluded zero vector.