= Axisymmetric linear map
{title2=$M=aI+cbb^T+d[b]_\times$}
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 $a+c\|b\|^2$. On the perpendicular plane its matrix in an oriented <orthonormal basis> is $\begin{pmatrix}a&-d\|b\|\\d\|b\|&a\end{pmatrix}$. Its eigenvalues are therefore $a+c\|b\|^2$ and $a\pm id\|b\|$, and
$$
\det M=(a+c\|b\|^2)(a^2+d^2\|b\|^2).
$$
To 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 $d\ne0$, only the axial eigenvalue is real; if $d=0$, the perpendicular plane also consists of real eigenvectors.
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