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### Eigenstructure of A

The defining property of the eigenvectors and eigenvalues of is the relation

which expands to

• The first element of is normalized arbitrarily to 1
• We have two equations in two unknowns and :

(We already know from above, but this analysis will find them by a different method.)

• Substitute the first into the second to eliminate :

• We have found both eigenvectors:

They are linearly independent provided and finite provided .

• The eigenvalues are then

• Assuming , they can be written as

• With , define , i.e., and .

• The eigenvalues become

as expected.

We again found the explicit formula for the frequency of oscillation:

where denotes the sampling rate. Or,

The coefficient range corresponds to frequencies .

We have shown that the example system oscillates sinusoidally at any desired digital frequency when , where denotes the sampling interval.

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