We will now use state-space analysisC.16 to determine Equations (C.130-C.133).
From Equations (C.125-C.129),
In matrix form, the state time-update can be written
A basic fact from linear algebra is that the determinant of a matrix is equal to the product of its eigenvalues. As a quick check, we find that the determinant of is
Note that . If we diagonalize this system to obtain , where diag , and is the matrix of eigenvectors of , then we have
where denotes the state vector in these new ``modal coordinates''. Since is diagonal, the modes are decoupled, and we can write
If this system is to generate a real sampled sinusoid at radian frequency , the eigenvalues and must be of the form
(in either order) where is real, and denotes the sampling interval in seconds.
Thus, we can determine the frequency of oscillation (and verify that the system actually oscillates) by determining the eigenvalues of . Note that, as a prerequisite, it will also be necessary to find two linearly independent eigenvectors of (columns of ).