= Solution
The kinetic energy of the three unit masses is $\frac12\sum_i\dot x_i^2$. Expanding the potential gives
$$
V=\frac12\left(3\sum_{i=1}^3x_i^2-2\sum_{i<j}x_ix_j\right).
$$
Thus the <small-oscillation mass and stiffness matrices> are
$$
\boxed{
T=I_3,\qquad
V=
\begin{pmatrix}
3&-1&-1\\
-1&3&-1\\
-1&-1&3
\end{pmatrix}
=4I_3-J,
}
$$
where $J$ is the all-ones matrix.
The vector $(1,1,1)^T$ is an <eigenvector> of the stiffness matrix with <eigenvalue> $1$. Every vector orthogonal to it has coordinate sum zero and is an eigenvector with eigenvalue $4$. Since $T=I$, the <generalized eigenvalue problem for small oscillations> gives $\omega^2$ equal to these eigenvalues. One orthonormal choice of <normal modes> is
$$
\begin{array}{c|c}
\text{mode}&\text{angular frequency}\\ \hline
\dfrac1{\sqrt3}(1,1,1)&1\\[4pt]
\dfrac1{\sqrt2}(1,-1,0)&2\\[4pt]
\dfrac1{\sqrt6}(1,1,-2)&2.
\end{array}
$$
The last two may be replaced by any orthonormal basis of the sum-zero plane because the frequency is degenerate. Hence \b[the normal frequencies are $\boxed{1,2,2}$.]
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