---
title: Complex Eigenvalues
module: Eigenvalues and Eigenvectors
moduleNumber: 5
lessonNumber: 5
order: 505
summary: >
  A real matrix with no real eigenvalues still has complex ones, occurring in
  conjugate pairs. A real 2-by-2 matrix with eigenvalue a plus b i is similar to
  a rotation-scaling matrix, whose rotation angle is the argument of the
  eigenvalue and whose scale factor is its modulus; the modulus decides whether
  the trajectories close up, spiral in, or spiral out.
topics: [Eigenvalues and Eigenvectors]
sources:
  - book: Lay
    ref: "§5.5 Complex Eigenvalues; Appendix B Complex Numbers"
---

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 $\mathbb{R}^n$; the complex roots instead encode a
rotation inside the real transformation. Extracting it requires letting $A$ act on
$\mathbb{C}^n$, the space of complex $n$-tuples.

## A matrix with no real eigenvalues

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

> **Worked example.** Show that $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ has
> no real eigenvalues, and find its complex ones.
>
> No nonzero real vector maps to a multiple of itself, and the characteristic equation
> $\lambda^2 + 1 = 0$ has roots $\lambda = i$ and $\lambda = -i$. Letting $A$ act on
> $\mathbb{C}^2$,
> $$
> A \begin{bmatrix} 1 \\ -i \end{bmatrix}
> = \begin{bmatrix} i \\ 1 \end{bmatrix}
> = i \begin{bmatrix} 1 \\ -i \end{bmatrix},
> $$
> so $i$ is an eigenvalue with eigenvector $(1, -i)$; likewise $-i$ has eigenvector
> $(1, i)$.

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

## Computing a complex eigenvector

> **Worked example.** Find the eigenvalues of
> $A = \begin{bmatrix} .5 & -.6 \\ .75 & 1.1 \end{bmatrix}$ and a basis for one
> eigenspace.
>
> The characteristic equation is $\lambda^2 - 1.6\lambda + 1 = 0$, so by the quadratic
> formula $\lambda = 0.8 \pm 0.6i$. For $\lambda = 0.8 - 0.6i$, the matrix
> $A - \lambda I$ is singular, so its two rows are complex multiples of each other and
> either row fixes the eigenvector. The second row gives
> $$
> 0.75\, x_1 = (-0.3 - 0.6i)\, x_2 \implies x_1 = (-0.4 - 0.8i)\, x_2.
> $$
> Choosing $x_2 = 5$ clears the decimals: $x_1 = -2 - 4i$, so
> $$
> v_1 = \begin{bmatrix} -2 - 4i \\ 5 \end{bmatrix}
> $$
> spans the eigenspace for $0.8 - 0.6i$.

## Conjugate pairs

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

> **Theorem (Conjugate eigenpairs).** If $A$ is a real matrix with complex eigenvalue
> $\lambda$ and corresponding eigenvector $x \in \mathbb{C}^n$, then $\bar\lambda$ is
> also an eigenvalue of $A$, with eigenvector $\bar{x}$. Complex eigenvalues of a real
> matrix occur in conjugate pairs.

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

$$
v_2 = \begin{bmatrix} -2 + 4i \\ 5 \end{bmatrix} = \bar{v}_1.
$$

$$
% caption: The conjugate eigenvalues of a real matrix are mirror images across the
% real axis; here $0.8 \pm 0.6i$ at radius $r = 1$ from the origin.
\begin{tikzpicture}[>=stealth, scale=1.6, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (1.5,0) node[right, black!70] {Re};
  \draw[->, black] (0,-1.0) -- (0,1.0) node[above, black!70] {Im};
  \draw[black] (0,0) circle (1);
  \draw[acc, dashed] (0,0) -- (0.8,0.6);
  \fill[acc] (0.8,0.6) circle (0.9pt) node[above right, acc] {$0.8 + 0.6i$};
  \fill[acc] (0.8,-0.6) circle (0.9pt) node[below right, acc] {$0.8 - 0.6i$};
  \node[black, anchor=west] at (0.62,0.28) {$r = 1$};
\end{tikzpicture}
$$

## The rotation-scaling building block

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

> **Rotation-scaling matrix.** Let $C = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$
> with $a, b$ real and not both zero. Its eigenvalues are $a \pm bi$, and with
> $r = \vert\lambda\vert = \sqrt{a^2 + b^2}$,
> $$
> C = r \begin{bmatrix} a/r & -b/r \\ b/r & a/r \end{bmatrix}
>   = \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix}
>     \begin{bmatrix} \cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi \end{bmatrix},
> $$
> where $\varphi$ is the angle from the positive real axis to $(a, b)$. Thus
> $x \mapsto Cx$ rotates by $\varphi$ and scales by $r$.

The angle $\varphi = \arg(a + bi)$ is the **argument** of the eigenvalue, and $r$ is
its modulus.

$$
% caption: The rotation-scaling matrix $C$ turns every vector by the argument
% $\varphi$ of its eigenvalue and stretches it by the modulus $r$.
\begin{tikzpicture}[>=stealth, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (3.2,0) node[right, black!70] {$x_1$};
  \draw[->, black] (0,-0.3) -- (0,2.6) node[above, black!70] {$x_2$};
  \draw[->, black, very thick] (0,0) -- (2,0.6) node[right] {$x$};
  \draw[->, acc, very thick] (0,0) -- (2.0,2.15) node[above right, acc] {$Cx$};
  \draw[black] (1.0,0.3) arc (16.7:47:1.05);
  \node[black] at (1.35,0.95) {rotate};
  \node[black] at (2.1,1.4) {scale by $r$};
\end{tikzpicture}
$$

## 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.

> **Theorem (Rotation-scaling factorization of a real 2×2 matrix).** Let $A$ be a real $2 \times 2$ matrix with complex eigenvalue
> $\lambda = a - bi$ ($b \neq 0$) and eigenvector $v \in \mathbb{C}^2$. Then
> $$
> A = P C P^{-1},
> \qquad
> P = \big[\; \operatorname{Re} v \;\; \operatorname{Im} v \;\big],
> \qquad
> C = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.
> $$

> **Worked example.** Factor $A = \begin{bmatrix} .5 & -.6 \\ .75 & 1.1 \end{bmatrix}$
> as $P C P^{-1}$, using $\lambda = 0.8 - 0.6i$ and
> $v_1 = \begin{bmatrix} -2 - 4i \\ 5 \end{bmatrix}$ from above.
>
> The real and imaginary parts of $v_1$ are the columns of $P$:
> $$
> P = \big[\, \operatorname{Re} v_1 \;\; \operatorname{Im} v_1 \,\big]
>   = \begin{bmatrix} -2 & -4 \\ 5 & 0 \end{bmatrix},
> \qquad
> C = P^{-1} A P = \begin{bmatrix} 0.8 & -0.6 \\ 0.6 & 0.8 \end{bmatrix}.
> $$
> Since $\vert\lambda\vert^2 = (0.8)^2 + (0.6)^2 = 1$, $C$ is a **pure rotation**.

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

$$
% caption: Iterating $x \mapsto Ax$ from a start point traces an ellipse, because the
% hidden rotation $C$ is viewed in the non-rectangular frame set by $P$.
\begin{tikzpicture}[>=stealth, scale=0.75, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.2,0) -- (3.2,0) node[right, black!70] {$x_1$};
  \draw[->, black] (0,-3.0) -- (0,3.0) node[above, black!70] {$x_2$};
  \draw[acc!45, thick, rotate=35] (0,0) ellipse (2.9 and 1.5);
  \foreach \a in {0,40,...,320} {
    \fill[acc] ({2.9*cos(\a)*cos(35) - 1.5*sin(\a)*sin(35)},{2.9*cos(\a)*sin(35) + 1.5*sin(\a)*cos(35)}) circle (2pt);
  }
  \node[acc] at (2.4,2.3) {elliptical orbit};
\end{tikzpicture}
$$

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 $x \mapsto Ax$ close
up, spiral in, or spiral out.

$$
% caption: With modulus below 1 the iterates spiral inward to the origin (an
% attractor); with modulus above 1 they spiral outward (a repeller).
\begin{tikzpicture}[>=stealth, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \definecolor{red}{HTML}{B0413E}
  \begin{scope}
    \draw[black] (-2,0) -- (2,0); \draw[black] (0,-2) -- (0,2);
    \draw[acc, thick] plot[domain=0:900,samples=140]
      ({(1.9*exp(-0.0018*\x))*cos(\x)},{(1.9*exp(-0.0018*\x))*sin(\x)});
    \fill[black] (0,0) circle (1.6pt);
    \node at (0,-2.5) {modulus $<$ 1: attractor};
  \end{scope}
  \begin{scope}[xshift=6cm]
    \draw[black] (-2,0) -- (2,0); \draw[black] (0,-2) -- (0,2);
    \draw[red, thick] plot[domain=0:900,samples=140]
      ({(0.18*exp(0.0026*\x))*cos(\x)},{(0.18*exp(0.0026*\x))*sin(\x)});
    \fill[black] (0,0) circle (1.6pt);
    \node at (0,-2.5) {modulus $>$ 1: repeller};
  \end{scope}
\end{tikzpicture}
$$

| $\vert\lambda\vert$ | Behavior of iterates $x, Ax, A^2 x, \dots$ | Origin is |
| --- | --- | --- |
| $\vert\lambda\vert = 1$ | closed elliptical orbit | a center |
| $\vert\lambda\vert < 1$ | spiral inward toward $0$ | an attractor |
| $\vert\lambda\vert > 1$ | 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,

$$
A = \begin{bmatrix} .8 & -.6 & 0 \\ .6 & .8 & 0 \\ 0 & 0 & 1.07 \end{bmatrix}
$$

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](/linear-algebra/eigenvalues/dynamical-systems).[^lay55]

[^lay55]: **Lay**, _Linear Algebra and Its Applications_, 5th ed., §5.5 — Complex Eigenvalues: eigenvalues and eigenvectors of a real matrix acting on $\mathbb{C}^n$, conjugate eigenpairs, the rotation-scaling matrix $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, and Theorem 9 factoring a real $2 \times 2$ matrix with complex eigenvalue as $P C P^{-1}$. Complex-number background from Appendix B.