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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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