The defining property of the eigenvectors and eigenvalues of is the relation
which expands to
(We already know from above, but this analysis will find them by a different method.)
They are linearly independent provided and finite provided .
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.