= Solution
With the same positive <displacement> convention for both masses, the first <eigenvector> equation is $(k+k_2-m\Omega^2)a_1-k_2a_2=0$. At the symmetric <frequency> it reduces to $k_2(a_1-a_2)=0$, while at the antisymmetric <frequency> it becomes $-k_2(a_1+a_2)=0$. Hence
$$
\boxed{\frac{a_1}{a_2}=+1\quad\text{for the symmetric mode},\qquad\frac{a_1}{a_2}=-1\quad\text{for the antisymmetric mode}}.
$$
In both <normal modes> the <displacement> magnitudes are equal. For the first, the masses move in <phase> and their separation stays unchanged; for the second they move in opposite <phase>, alternately approaching and separating. A negative signed <amplitude> ratio records a <phase> difference of $\pi$, not a negative <displacement> magnitude. The left panel of the later figure plots these signed <normal mode> shapes with the fixed wall values included.
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