= Solution
A massless <eigenvector> must lie in the <null space> of $R$, so its coefficients satisfy $c_1=qc_2$ and $c_2=qc_3$. Hence they are proportional to $(q^2,q,1)$. The original gauge <kinetic terms> have the Euclidean unit <matrix> in species space, so normalize this vector by its ordinary squared length. The massless <field> is
$$
\boxed{A_1^\mu=\frac{q^2B_1^\mu+qB_2^\mu+B_3^\mu}{\sqrt{q^4+q^2+1}}.}
$$
The remaining two <eigenvectors> can be chosen orthogonal to it, making the full transformation orthogonal and preserving canonical vector <kinetic terms>. At $q=1$, the formula is exactly $(B_1^\mu+B_2^\mu+B_3^\mu)/\sqrt3$ as required. This null direction and its normalization are unaffected by the common mass-scale correction in part b.
For $q>1$, the massless mode has geometrically unequal components; a matter <field> coupled to one end can therefore couple more weakly to this surviving vector than a <field> coupled to the other end. This follows directly from the <eigenvector> coefficients, without changing the equal microscopic gauge <coupling> $g$.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2017-abelian-gauge-chain-spectrum.png]
{title=Massless-vector profile and canonically normalized masses of the <three-site Abelian Higgs chain>}
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