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### Eigenstructure

Starting with the defining equation for an eigenvector and its corresponding eigenvalue ,

we get, using Eq.(C.158),

 (C.161)

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.162) gives us two equations in two unknowns:

 (C.162) (C.163)

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.163) to find the eigenvalues:

Subsections
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