Axisymmetric linear map ($M=aI+cb{b}^T+d[b{]}_{\times }$)

For a fixed nonzero real vector $b$ in three dimensions, a real linear map commuting with every proper rotation about that axis has the displayed form, where $[b{]}_{\times }x=b\times x$ is the cross-product matrix. On the axis it acts by $a+c\Vert b{\Vert }^2$. On the perpendicular plane its matrix in an oriented orthonormal basis is $\left(\begin{array}{cc}a& -d\Vert b\Vert \\ d\Vert b\Vert & a\end{array}\right)$. Its eigenvalues are therefore $a+c\Vert b{\Vert }^2$ and $a\pm id\Vert b\Vert$, andTo see the converse, commuting with all rotations preserves their common fixed axis and its perpendicular plane; there can be no nonzero coupling vector fixed by every planar rotation. A planar matrix commuting with every rotation is a scalar identity plus a scalar right-angle rotation. Thus the block description is exhaustive. If $d\ne 0$, only the axial eigenvalue is real; if $d=0$, the perpendicular plane also consists of real eigenvectors.