Example

For the vibrating mass-spring system introduced in §2.1 and Figure 2.1, we assume that

1. all masses $m_i=1$, so $M=I$, and
2. all damping constants $b_i=0$, so $B=0$.

This simplies the equations of motion to $\ddot{x}(t) = -K x(t)$. We solve them by substituting $x(t) = e^{\lambda t} x$, where $x$ is a constant vector and $\lambda$ is a constant scalar to be determined. This yields

$$Kx = -\lambda^2 x.$$

Thus $x$ is an eigenvector and $-\lambda^2$ is an eigenvalue of the symmetric positive definite tridiagonal matrix $K$. Thus $\lambda$ is pure imaginary and we get that $x(t)$ is periodic with period $2 \pi/ \vert\lambda\vert$. Symmetric tridiagonal matrices have particularly fast and efficient eigenvalue algorithms.

Later sections deal with the cases of nonunit masses $m_i$ and nonzero damping constants $b_i$.