Starting with the defining equation for an eigenvector
and its
corresponding eigenvalue
,
we get, using Eq.(C.158),
Equation (C.162) gives us two equations in two unknowns:
As
approaches
(no damping), we obtain
Thus, we have found both eigenvectors:
They are linearly independent provided
. In the undamped
case (
), this holds whenever
. The eigenvectors are
finite when
. Thus, the nominal range for
is the
interval
.
We can now use Eq.(C.163) to find the eigenvalues: