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Next: Generalized Non-Hermitian Eigenproblems   Up: Non-Hermitian Eigenproblems  J. Demmel Previous: Related Eigenproblems   Contents   Index


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,


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

\begin{displaymath}Ay = \lambda y.\end{displaymath}

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

Susan Blackford 2000-11-20