Example

We continue to use the example introduced in §2.1 and Figure 2.1. We now consider the case where there are arbitrary positive masses $m_i>0$, but the damping constants $b_i$ are zero. This simplifies the equations of motion to $M \ddot{x}(t) = -K x(t)$. We again 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

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

Thus $x$ is an eigenvector and $-\lambda^2$ is an eigenvalue of the generalized Hermitian eigenproblem $Kx = \mu Mx$. Since $K$ and $M$ are positive definite, the eigenvalues $-\lambda^2$ are positive, so $\lambda$ is pure imaginary and we find that $x(t)$ is periodic with period $2 \pi/ \vert\lambda\vert$.

Following item 3 in §2.3.7, we may convert this to a standard Hermitian eigenvalue problem as follows. Let $M = LL^T$ be the Cholesky decomposition of $M$. Thus $L$ is simply the diagonal matrix ${\rm diag}( m_1^{1/2} ,\ldots, m_n^{1/2} )$. Then the eigenvalues of $Kx = \mu Mx$ are the same as the eigenvalues of the symmetric tridiagonal matrix

$$\hat{K} = L^{-1} K L^{-T} = \bmat{ccccc} \frac{k_1 + k_2}{m... ...{n-1}m_n}} & \frac{k_n}{m_n} \emat. \vspace{\belowdisplayskip}$$ (2)

Symmetric tridiagonal matrices have particularly fast and efficient eigenvalue algorithms.