Chapter 6.P2, Problem 8P

A symmetric matrix $G$ is said to be negative definite if $x^{T}Gx<0$ for every $x\ne 0$. Prove that a symmetric matrix $G$ is negative definite if and only if its eigenvalues are all negative. Use this result to show that $G$ in (3) has negative eigenvalues.

$\text{G}={\omega }_0{}^2\left(\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -2\end{array}\right)u(t)=\left(\begin{array}{c}{u}_1(t)\\ u_2(t)\\ u_3(t)\end{array}\right)=\frac{1}{m}\left(\begin{array}{c}{\hat{u}}_1(t)\\ {\hat{u}}_2(t)\\ {\hat{u}}_3(t)\end{array}\right)$(3)