= Solution
Put $\boldsymbol x=q e^{i\omega t}$ in <11/a/solution>. The <normal mode> equation is $Aq=\lambda q$, with $\lambda=m\omega^2/k$. Its <characteristic polynomial> is
$$
\det(A-\lambda I)=(2-\lambda)\big[(2-\lambda)^2-2\big].
$$
The three <eigenvalues> are $2-\sqrt2$, $2$, and $2+\sqrt2$. Solving the corresponding linear equations yields
$$
\boxed{\begin{array}{c|c|c}
\text{mode}&\omega&\text{mode shape }q\\\hline
1&\sqrt{(2-\sqrt2)k/m}&(1,\sqrt2,1)^T\\
2&\sqrt{2k/m}&(1,0,-1)^T\\
3&\sqrt{(2+\sqrt2)k/m}&(1,-\sqrt2,1)^T
\end{array}}
$$
Each <mode shape> may be multiplied by any nonzero scalar. Their squared <Euclidean norms> are $4,2,4$, respectively, so unit-normalized <eigenvectors> are $q_1/2,q_2/\sqrt2,q_3/2$. They are mutually orthogonal, as expected for the real symmetric <stiffness matrix> with equal <masses>.
The entries in the table are <angular frequencies>; the corresponding ordinary <frequencies> in cycles per unit time are $f_j=\omega_j/(2\pi)$. All three are positive because the fixed walls remove uniform translation as a free motion.
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