Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 3 / 9C / b / Solution

Let $R_{\theta }$ be rotation about the third axis, so invariance means $R_{\theta }T{R}_{\theta }^T=T$. At $\theta =\pi$, its diagonal signs are $(-1,-1,1)$; each mixed entry $T_{13},T_{23},T_{31},T_{32}$ changes sign and must vanish.

For the remaining planar block $M=\left(\begin{array}{cc}p& q\\ r& s\end{array}\right)$, the quarter-turn $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ givesHence $s=p$ and $r=-q$. Writing the constants as $\alpha ,\omega ,\beta$ givesConversely, $M=\alpha I-\omega J$ commutes with every planar rotation, so every displayed matrix is invariant under every rotation about the axis. This is an oriented-axis invariant Cartesian tensor. Symmetry was not assumed, so the antisymmetric parameter $\omega$ need not vanish.