= Solution
Expanding the <determinant> along the first row gives
$$
\det(D-\lambda I)=(2-\lambda)\bigl[(2-\lambda)^2-2\bigr].
$$
The <eigenvalues> and normalized spatial <eigenvectors>, in increasing frequency order, are
$$
\boxed{\begin{array}{c|c|c}
\lambda_j&\mathbf a^{(j)}&\omega_j^2\\\hline
2-\sqrt2&\tfrac12(1,\sqrt2,1)&(k/m)(2-\sqrt2)\\
2&\tfrac1{\sqrt2}(1,0,-1)&2k/m\\
2+\sqrt2&\tfrac12(1,-\sqrt2,1)&(k/m)(2+\sqrt2)
\end{array}}.
$$
For each mode both signs of $\omega_j$ give the same <cosine> displacement. An arbitrary overall mode amplitude remains free. These vectors are the $N=3$ instances of the <sine eigenvectors of the fixed-end spring chain>.
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