Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 7 / a / iii / Solution

Using the three real eigenvectors and positive frequencies in part (ii), the general motion isThere are 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.