Complex Eigenvalues

The characteristic equation of an $n \times n$ real matrix has degree $n$ , so it has $n$ roots once complex numbers are allowed. When some of those roots are not real, the matrix has no eigenvectors in $R^{n}$ ; the complex roots instead encode a rotation inside the real transformation. Extracting it requires letting $A$ act on $C^{n}$ , the space of complex $n$ -tuples.

A matrix with no real eigenvalues

A quarter-turn rotation is the simplest matrix with no real eigenvalues.

The real matrix theory carries over verbatim: a complex scalar $λ$ satisfies $det (A - λ I) = 0$ if and only if there is a nonzero $x \in C^{n}$ with $A x = λ x$ .

Computing a complex eigenvector

Conjugate pairs

Complex eigenvalues of a real matrix never appear alone. Because $A$ is real, $\overline{A x} = A \overset{x}{ˉ}$ , so applying conjugation to $A x = λ x$ gives $A \overset{x}{ˉ} = \overset{ˉ}{λ} \overset{x}{ˉ}$ .

For the matrix above, the second eigenvalue $0.8 + 0.6 i$ has eigenvector

v_{2} = [- 2 + 4 i 5] = \overset{v}{ˉ}_{1} .

The conjugate eigenvalues of a real matrix are mirror images across the real axis; here

0.8 \pm 0.6 i

at radius

r = 1

from the origin.

The rotation-scaling building block

A real $2 \times 2$ matrix of the following shape acts as a rotation combined with a uniform scaling.

The angle $φ = ar g (a + bi)$ is the argument of the eigenvalue, and $r$ is its modulus.

The rotation-scaling matrix

C

turns every vector by the argument

φ

of its eigenvalue and stretches it by the modulus

r

The rotation inside a real matrix

Any real $2 \times 2$ matrix with a complex eigenvalue is similar to a rotation-scaling matrix. The change of basis is built from one complex eigenvector.

The factorization $A = P C P^{- 1}$ reads: change variables by $P^{- 1}$ , rotate by $C$ , change back. The transformation $x \mapsto A x$ is a rotation seen through the skewed, unequally scaled coordinate system whose axes are the columns of $P$ .

Iterating

x \mapsto A x

from a start point traces an ellipse, because the hidden rotation

C

is viewed in the non-rectangular frame set by

P

The orbit is an ellipse rather than a circle exactly because the columns of $P$ are not orthonormal: in the $x$ -coordinates the circular motion of $C$ is stretched along the eigenvector directions.

Modulus governs the spiral

The modulus of the eigenvalue determines whether the iterates of $x \mapsto A x$ close up, spiral in, or spiral out.

With modulus below 1 the iterates spiral inward to the origin (an attractor); with modulus above 1 they spiral outward (a repeller).

$∣ λ ∣$	Behavior of iterates $x, A x, A^{2} x, \dots$	Origin is
$∣ λ ∣ = 1$	closed elliptical orbit	a center
$∣ λ ∣ < 1$	spiral inward toward $0$	an attractor
$∣ λ ∣ > 1$	spiral outward, unbounded	a repeller

The same rotation persists in higher dimensions. If a $3 \times 3$ real matrix has a complex eigenvalue, there is a two-dimensional invariant plane on which $A$ acts as a rotation, possibly with scaling, while the remaining real eigenvalue governs motion transverse to that plane. For instance,

A = .8 .6 0 - .6 .8 0 00 1.07

rotates the $x_{1} x_{2}$ -plane and stretches the $x_{3}$ -direction by $1.07$ ; a generic trajectory spirals in the plane while drifting away along the third axis.

The dynamical readings of these pictures (attractor, repeller, spiral point) are formalized for both discrete and continuous systems under dynamical systems.¹

Lay, Linear Algebra and Its Applications, 5th ed., §5.5 — Complex Eigenvalues: eigenvalues and eigenvectors of a real matrix acting on $C^{n}$ , conjugate eigenpairs, the rotation-scaling matrix $[a b - b a]$ , and Theorem 9 factoring a real $2 \times 2$ matrix with complex eigenvalue as $P C P^{- 1}$ . Complex-number background from Appendix B. ↩