Chapter 6.P2, Problem 2P

Find the eigenvalues and eigenvectors of the matrix $G$ in (3). Assume that ${\omega }_0{}^2=1$. Plot the components of the eigenvectors. Verbally, and with the aid of sketches, describe the normal modes of vibration in terms of the motion of the masses relative to one another. $G={\omega }_0{}^2\left(\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -2\end{array}\right),u(t)=\left(\begin{array}{c}{u}_1(t)\\ u_2(t)\\ u_3(t)\end{array}\right)=\frac{1}{m}\left(\begin{array}{c}{\hat{u}}_1(t)\\ {\hat{u}}_2(t)\\ {\hat{u}}_3(t)\end{array}\right)$ (3)