Complex Eigenvalues — Question 8

Question 8 – Solution

Strategy. Sampling observes angles only modulo complete revolutions; additional information is needed to recover the continuous frequency.

Step 1: Solve and impose integer samples. The eigenvalues are $-1\pm i\omega$, and $$\boxed{X(t)=e^{-t}\binom{\cos\omega t}{\sin\omega t}.}$$ The measurement at $n=1$ requires $\omega=2\pi k$, $k\in\mathbb Z$. Every such value satisfies all integer samples; the nonzero-frequency assumption excludes $k=0$.

Step 2: Identify what the samples conceal. The sign of $k$ gives counterclockwise or clockwise rotation, and $|k|$ gives revolutions per unit time. Neither is fixed by the integer samples. For every permitted nonzero frequency the continuous angle changes and the radius decays, tracing a spiral. The sampled points all lie on one ray because whole turns occur between them, not because that ray is an invariant trajectory.

Step 3: Use the quarter-time observation. The additional angle condition is $\omega/4=\pi/2+2\pi m$. Thus $$\boxed{\omega=2\pi(1+4m),\qquad m\in\mathbb Z.}$$ These values already satisfy the integer constraints. Negative possibilities remain, for example $m=-1$ gives $\omega=-6\pi$. No zero frequency occurs in this family.

Step 4: Use the external frequency bound. Under $0<\omega<4\pi$, only $m=0$ remains, giving $\boxed{\omega=2\pi}$. Without that bound, $2\pi$ and $-6\pi$ both fit all stated observations, despite opposite directions and different speeds. The figure plots $e^t y(t)$ for these two solutions: they agree at the recorded times but differ between them. The vertical variable is the rescaled component, not the radius or a phase-plane coordinate.