Complex Eigenvalues

Not all matrices have real eigenvectors. When the characteristic polynomial has no real roots, the eigenvalues come in complex conjugate pairs — and the geometric behavior becomes rotational rather than stretching.

When eigenvalues are complex

If the discriminant $\Delta = \text{tr}(A)^2 - 4\det(A) < 0$, the eigenvalues are:

These are complex conjugates: $\lambda = a \pm bi$. The real part a controls growth/decay; the imaginary part b controls rotation.

The modulus determines stability

The modulus (magnitude) of the complex eigenvalue is:

Apply the matrix repeatedly — points orbit the origin:

• $|\lambda| = 1$: pure rotation, orbits stay on a circle
• $|\lambda| > 1$: spiral outward (unstable)
• $|\lambda| < 1$: spiral inward, converges to zero (stable)

The rotation angle

The angle of rotation per step is $\theta = \arctan(b/a)$ — the argument of the complex eigenvalue. A rotation matrix of angle θ has eigenvalues$e^{\pm i\theta} = \cos\theta \pm i\sin\theta$.

Try it

Pick a preset and watch how points spiral (or orbit) under repeated application of the matrix. The dashed circle is the starting unit circle. Use the steps slider to see long-term behavior.

import numpy as np

A = np.array([[0.8, -0.4], [0.4, 0.8]])
eigenvalues = np.linalg.eigvals(A)

for lam in eigenvalues:
    print(f"λ = {lam:.3f}, |λ| = {abs(lam):.3f}")
# λ = 0.800+0.400j, |λ| = 0.894 → stable (spirals in)