Complex Eigenvalues

A complex eigenvalue is simply what the characteristic equation hands you when a matrix turns every direction, so no real vector stays on its own line. "Complex" here doesn't mean complicated. It means made of two parts: a real part $a$ and an imaginary part $b$ (the number multiplying $i = \sqrt{-1}$). They're useful because those two parts tell you directly how much the matrix rotates and how much it scales. All we're really saying in the formula is: $a$ and $b$ (the two parts of the eigenvalue $a + bi$), $r = \sqrt{a^2 + b^2}$ (the eigenvalue's size, which is how much the matrix scales) and the matrix of $\cos\varphi$ and $\sin\varphi$ (a pure rotation by $\varphi$, the angle of $a + bi$). So $C$ is just "scale by $r$" times "rotate by $\varphi$"; bear with it.

Take a look at the rotation from last lesson. It had no real eigenvectors, because every direction turns, and its characteristic equation $\lambda^2 + 1 = 0$ had no real roots. The roots were still there, though, just complex: $\pm i$. That's the general pattern. Complex eigenvalues of real matrices come in conjugate pairs, because the characteristic polynomial has real coefficients, so if $a + bi$ is a root, so is $a - bi$. Every $2\times2$ real matrix with eigenvalues $a \pm bi$ is similar to the rotation–scaling matrix $C$ above.

Repeated application: each step multiplies the size by $r$ and adds another $\varphi$ to the angle, so $C^k$ rotates by $k\varphi$ and scales by $r^k$. With $r < 1$ points spiral in; with $r > 1$ they spiral out; with $r = 1$ they circle forever.

A common mistake. It's tempting to think the scale factor is $a$, because $a$ is what sits on the diagonal. It isn't: the scale factor is $r = \sqrt{a^2 + b^2}$, which uses both parts. For example, $\begin{bmatrix}0.8 & -0.6\\ 0.6 & 0.8\end{bmatrix}$ has $0.8$ on the diagonal, but $r = 1$, so it doesn't shrink anything; it just rotates. The angle needs both parts too: $\cos\varphi = a/r$ and $\sin\varphi = b/r$.

There's an animation further down that shows the points spiraling in step by step.

Oscillating systems (springs, circuits, predator–prey cycles) have complex eigenvalues; in graphics, rotation–scaling matrices animate spirals and orbits.