Complex Eigenvalues

Complex eigenvalues are simply eigenvalues that come out with an $i$ in them, which happens when the characteristic equation has no real roots. "Complex" doesn't mean complicated here; it means the number has a real part and an imaginary part. They're good news for understanding: they tell you the system rotates or oscillates. The form $\lambda = \alpha \pm \beta i$ looks abstract, but it's just two numbers: $\beta$ (how fast it turns, in radians per unit time) and $\alpha$ (whether it grows, $\alpha > 0$, or shrinks, $\alpha < 0$, as it turns). That's really all there is to read off.

Where the rotation comes from. You saw this with second-order equations: complex roots $\alpha \pm \beta i$ gave $e^{\alpha t}\cos\beta t$ and $e^{\alpha t}\sin\beta t$. The same thing happens here. By Euler's formula, $e^{(\alpha + \beta i)t} = e^{\alpha t}(\cos\beta t + i\sin\beta t)$, so the $e^{\alpha t}$ part scales and the $\cos$ and $\sin$ part turns.

Real solutions. Take one complex eigenvalue and eigenvector, form $e^{(\alpha + \beta i)t}\vec v$, and expand with Euler's formula. Its real part and imaginary part are two independent real solutions. Think about why that's allowed: $A$ has real entries, so if a complex vector solves $\vec x' = A\vec x$, its real and imaginary parts each have to solve it separately. You don't need the other eigenvalue $\alpha - \beta i$; it just gives the same two solutions again.

Reading the picture: $\alpha < 0$ spirals in (stable), $\alpha > 0$ spirals out, $\alpha = 0$ gives closed ellipses (a center). The period of rotation is $\frac{2\pi}{\beta}$.

A common mistake. It's tempting to think the period depends on the whole eigenvalue, maybe $\frac{2\pi}{|\lambda|}$, because $\lambda$ is the number you computed. It isn't: only the imaginary part sets the turning, so the period is $\frac{2\pi}{\beta}$. The real part $\alpha$ only changes the size of the spiral, not how long each lap takes.

Predator–prey cycles, the oscillating currents in LC circuits and the swaying of a building in wind are rotations in phase space caused by complex eigenvalues.