Example

We continue to use the example introduced in §2.1 and Figure 2.1. We now consider the case of unit masses $m_i=1$ but nonzero damping constants $b_i$. (The case of nonunit masses is handled in §2.6.8.) This simplifies the equations of motion to $\ddot{x}(t) = -B \dot{x} (t) -K x(t)$. We solve them by changing variables to

$$y(t) = \bmat{c} \dot{x}(t) \\ x(t) \emat,$$

yielding

We again solve by substituting $y(t) = e^{\lambda t} y$, where $y$ is a constant vector and $\lambda$ is a constant scalar to be determined. This yields

$$Ay = \lambda y.$$

Thus $y$ is an eigenvector and $\lambda$ is an eigenvalue of the non-Hermitian matrix $A$.