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Complex Eigenvalues and Rotation

When the eigenvalues come out complex, the eigenvalue method does not break — it starts to spin. A complex eigenvalue a + bi hides a rotation, and turning one complex solution into two real ones reveals trajectories that spiral or loop instead of running along straight lines.

The moment the eigenvalues turn complex

In the previous guide every eigenvalue was a real number, and each one handed you a straight-line solution: a direction in the phase plane along which the motion was pure growth or pure decay, e^(lambda t) times a fixed eigenvector direction. But the characteristic equation of a 2x2 system is a quadratic, and quadratics do not always have real roots. The discriminant can go negative, and then the eigenvalue method delivers a complex eigenvalue of the form lambda = a + bi.

For a real matrix A this never happens in isolation. Complex roots of a real polynomial always arrive in conjugate pairs, so if a + bi is an eigenvalue then a - bi is the other one automatically. Their eigenvectors are conjugates too: solve (A - lambda I) v = 0 for lambda = a + bi and you get a vector v with complex entries; the vector for a - bi is just its entry-by-entry conjugate. There are no real eigenvectors at all here, which is the whole point — there is no real direction the system simply slides along, because the system is turning.

Euler's formula is the rotation inside the exponential

Everything turns on one identity you have met before: Euler's formula, e^(i*theta) = cos(theta) + i*sin(theta). Read it slowly. The left side is an exponential — the same growth-machine that gave straight-line solutions for real eigenvalues. The right side is pure rotation, a point walking around the unit circle. So a purely imaginary exponent does not grow or shrink anything; it spins. This is the seed of every spiral and every loop you are about to see.

Now split the complex eigenvalue into its parts. The factor e^((a+bi)t) = e^(at) * e^(i*bt), and by Euler's formula the second piece is cos(bt) + i*sin(bt). The eigenvalue's real part a sits in e^(at) and controls size — whether trajectories swell outward or shrink inward. Its imaginary part b sits in cos(bt) and sin(bt) and controls turning — how fast the system sweeps around. One number sets the breathing, the other sets the spinning, and they act at the same time without interfering.

This is the exact system-level echo of something you already saw for a single second-order equation. There, the complex conjugate roots r = a +/- bi of the characteristic equation gave solutions e^(at) cos(bt) and e^(at) sin(bt) — a decaying or growing oscillation with amplitude and phase. The same a and b, the same envelope-times-oscillation shape. A system of two first-order equations is just that one story told in two coordinates at once, so the rotation that was invisible in a single y(t) becomes a literal turning in the (x, y) plane.

From one complex solution to two real ones

Here is the trick that makes the whole complex eigenvalue case practical, and it rests on a small but powerful fact: when the matrix is real, the real part and the imaginary part of any complex solution are EACH, on their own, a real solution. You therefore do not need both members of the conjugate pair. Take just one — say the solution built from lambda = a + bi — expand it out, and harvest its real and imaginary parts as two separate, linearly independent real solutions.

  1. Solve (A - lambda I) v = 0 for just ONE eigenvalue lambda = a + bi to get a single complex eigenvector v. Ignore the conjugate eigenvalue entirely — it will give you nothing new.
  2. Form the one complex solution z(t) = e^((a+bi)t) v, and rewrite the scalar exponential as e^(at)(cos(bt) + i*sin(bt)) using Euler's formula.
  3. Multiply everything out and separate z(t) into z(t) = x_re(t) + i * x_im(t), gathering all the real terms into x_re and all the i-attached terms into x_im.
  4. The two real vector functions x_re(t) and x_im(t) are your real fundamental pair. The general real solution is x(t) = C1 * x_re(t) + C2 * x_im(t), with C1 and C2 fixed by the initial condition.

What the trajectories actually look like

Now picture it in the phase plane. Because every solution is e^(at) times a turning vector, the geometry is read straight off the two numbers a and b. The factor e^(at) sets the radius and the oscillation sets the angle, so the trajectory is some flavour of spiral or circle. There are exactly three pictures, and the sign of the real part a decides which one you get.

  real part a          picture in the (x,y) plane     stability
  -----------------    -------------------------       ---------------------
  a < 0  (negative)    spiral winding INWARD           asymptotically stable
  a = 0  (zero)        closed loops (ellipses)         a center: neutrally stable
  a > 0  (positive)    spiral winding OUTWARD          unstable

  the imaginary part b only sets HOW FAST it turns (angular speed)
  and which way it winds; it never changes inward vs. outward.
The three complex-eigenvalue portraits. The real part a alone decides in/out; the imaginary part b only sets the rate and direction of turning.

When a is negative the solution spirals inward to the origin and you have a stable spiral point — think of a swinging door with friction, looping as it settles to rest. When a is positive the same spiral runs in reverse, flinging outward, and the origin is an unstable spiral. The borderline a = 0 is the cleanest and the strangest: the exponential factor is e^(0) = 1, the radius never changes, and the trajectories are closed loops that circle forever. That equilibrium is a center — perpetual orbiting with neither gain nor loss, the phase-plane face of an undamped oscillation.

Honest edges: which way, and why the center is fragile

Two honest caveats keep you from over-trusting the picture. First, the eigenvalues tell you that the motion rotates and at what rate, but NOT which way around. The sense of rotation — clockwise or counter-clockwise — depends on the actual entries of A, not on the eigenvalues alone. The reliable way to settle it is to plant one arrow: pick a single point, compute the velocity vector A x there, and see which way the flow points. The eigenvalues give you the shape; one sample arrow gives you the orientation.

Second, and more important, the center (a = 0) is a knife-edge that survives only in idealized linear systems. Its closed loops are infinitely fragile: nudge the real part a even slightly positive or negative — by adding the faintest friction, or by the tiny corrections that any real nonlinear model carries — and the loops immediately unwind into an inward or outward spiral. This is exactly the borderline that linearization (Hartman-Grobman) refuses to certify. The theorem only guarantees the linear picture near a hyperbolic equilibrium, one whose eigenvalues have NONzero real part; a pure center has a = 0, sits off that guarantee, and its true behaviour in the full nonlinear system must be checked by other means.

With this guide you have closed the gap that real distinct eigenvalues left open. Together those two cases cover every situation where the eigenvalues are distinct — whether real (straight lines) or complex (spirals and loops). What remains is the awkward leftover: when the discriminant is exactly zero and the two eigenvalues collapse into one. A single eigenvalue may not supply enough independent eigenvectors to fill the plane, and patching that hole is the job of generalized eigenvectors in the final guide of this rung.