Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 11 / c / Solution

For $\lambda =1$, the first eigenvector equation gives $y_1=y_2$. For $\lambda =4$, it gives $y_1=-2y_2$. ThereforeThe lower mode moves the masses in phase with equal displacements. In the higher mode they move in opposite directions, with the lighter mass's amplitude twice that of the heavier mass. Their eigenvectors satisfy mass-weighted orthogonality with respect to the physical mass matrix $\mathrm{diag}(m,2m)$, although their ordinary Euclidean dot product is nonzero.