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Eigenstructure
Starting with the defining equation for an eigenvector
and its
corresponding eigenvalue
,
we get, using Eq.
(C.134),

(C.137) 
We normalized the first element of
to 1 since
is an
eigenvector whenever
is. (If there is a missing solution
because its first element happens to be zero, we can repeat the
analysis normalizing the second element to 1 instead.)
Equation (C.138) gives us two equations in two unknowns:
Substituting the first into the second to eliminate
, we get
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.139) to find the eigenvalues:
Subsections
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