= Solution
Using the three real <eigenvectors> and positive frequencies in part (ii), the general motion is
$$
\boxed{\mathbf x(t)=\sum_{j=1}^3\mathbf u_j[A_j\cos(\omega_jt)+B_j\sin(\omega_jt)].}
$$
There are \b[six independent real constants], $A_1,A_2,A_3,B_1,B_2,B_3$, corresponding to the three initial displacements and three initial velocities. With the displayed orthonormal <eigenvectors>, $A_j=\mathbf u_j^T\mathbf x(0)$ and $B_j=\mathbf u_j^T\dot{\mathbf x}(0)/\omega_j$. This spans all solutions because the normal-mode <eigenvectors> form a basis.
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