= Solution
For a <unit vector>, the radial coordinate is $r=1$ and the <spherical coordinates> give
$$
(n_x,n_y,n_z)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).
$$
The <Pauli matrices> therefore give
$$
\boxed{H=\begin{pmatrix}\cos\theta&e^{-i\phi}\sin\theta\\e^{i\phi}\sin\theta&-\cos\theta\end{pmatrix}.}
$$
The diagonal entries are real and the off-diagonal entries are <complex conjugates>, so $H^*=H$. Its trace is zero and its <determinant> is $-1$, giving <eigenvalues> $+1$ and $-1$. A convenient <orthonormal eigenbasis> is
$$
\boxed{v_+=\binom{\cos(\theta/2)}{e^{i\phi}\sin(\theta/2)},\qquad
v_-=\binom{-\sin(\theta/2)}{e^{i\phi}\cos(\theta/2)}.}
$$
The half-angle identities verify $Hv_\pm=\pm v_\pm$, $v_\pm^*v_\pm=1$ and $v_+^*v_-=0$. Multiplication of either vector by any unit-modulus phase gives an equivalent normalized <eigenvector>. At a pole, the azimuth is undefined geometrically, but the remaining phase freedom makes the chosen expressions consistent with the same one-dimensional <eigenspaces>.
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