= Solution
Write $d=2k-m\omega^2$. The characteristic <determinant> of the tridiagonal matrix is
$$
\det(V-\omega^2T)=d(d^2-k^2)-k^2d=d(d^2-2k^2).
$$
The three values of $d$ are $\sqrt2k$, $0$ and $-\sqrt2k$. Therefore the <normal mode> frequencies and convenient Euclidean-normalized generalized <eigenvectors> are
$$
\boxed{\begin{array}{c|c}
\omega^2&\mathbf u\\\hline
(2-\sqrt2)k/m&(1,\sqrt2,1)^T/2\\
2k/m&(1,0,-1)^T/\sqrt2\\
(2+\sqrt2)k/m&(1,-\sqrt2,1)^T/2
\end{array}.}
$$
Each pair satisfies $V\mathbf u=\omega^2T\mathbf u$. The first mode moves all masses in phase, the middle mode leaves the central mass fixed, and the last has the central mass moving opposite to the outer masses. All squared frequencies are positive, consistent with the positive spring potential and fixed endpoints.
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