= Axisymmetric symmetric rank-two tensor
{title2=$T=\alpha I+\beta yy^T$}
A <symmetric tensor> invariant under rotations about a fixed nonzero vector $y$ has the form $\alpha I+\beta yy^T$. Choose the third coordinate axis parallel to $y$: rotation by a half-turn makes the mixed axial-transverse entries vanish, and all transverse rotations make the transverse block a scalar multiple of the identity. Its two transverse eigenvalues are $\alpha$ and its axial eigenvalue is $\alpha+\beta|y|^2$. The <trace> and the quadratic contraction $y^TTy$ determine these two coefficients. Full isotropy holds exactly when $\beta=0$.
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