Complex Numbers in One Page

Complex numbers appear in this module for exactly one reason, and it is worth stating up front so you know how much of this page you need.

A matrix full of real numbers can have eigenvalues that are not real. A rotation is the standard example: it turns every vector, so there is no direction it merely stretches, so there is no real eigenvector — and yet the characteristic polynomial still has roots, because polynomials always do once you allow complex numbers. §4.2 hits this on its second example, and a reader who has never seen $i$ stalls there.

That is the whole motivation. This page covers what you need for it and stops.

What you’ll learn

• What $i$ is, and why it was invented.
• The complex plane, and why multiplication by a complex number is a rotation-and-scale.
• Modulus and argument, and Euler’s formula tying them to $\cos$ and $\sin$.
• Conjugates, and why complex eigenvalues of a real matrix always come in pairs.
• The precise reason a rotation matrix has no real eigenvalues.

Intuition: a number that turns things

Real numbers live on a line. Multiplying by a positive real stretches along that line; multiplying by $-1$ flips it — a rotation by $180°$.

So ask: what would rotate by $90°$? Doing it twice must give the $180°$ flip, so it must be a number whose square is $-1$. No real number qualifies, so we name one:

$$i^2 = -1$$

Read $i$ as “the quarter turn” rather than “the square root of minus one”. That reading makes everything below obvious instead of mysterious, and it is exactly the reading that connects complex numbers to rotation matrices.

diagram Diagram mermaid

  flowchart LR
  R["multiply by a positive real
stretch along the line"] --> P["the real line is not enough"]
  N["multiply by -1
rotate by 180 degrees"] --> P
  P --> Q["what rotates by 90 degrees?
its square must be -1"]
  Q --> I["call it i
now numbers live on a plane"]
  I --> M["multiplying by a+bi
= scale by modulus, rotate by argument"]

The math

The plane

A complex number is a pair of reals wearing one symbol:

$$z = a + bi, \qquad a = \operatorname{Re}(z),\quad b = \operatorname{Im}(z)$$

Plot $a$ horizontally and $b$ vertically and you have the complex plane. A complex number is a point in $\mathbb{R}^2$ — with one extra piece of structure real pairs do not have: a way to multiply two of them together and get a third.

Arithmetic

Addition is componentwise, exactly like vectors:

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

Multiplication is where the structure is. Expand and use $i^2 = -1$:

$$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$$

The minus sign in $ac - bd$ is the entire content of $i^2 = -1$, and it is the reason multiplication rotates.

Modulus and argument

$$\lvert z\rvert = \sqrt{a^2 + b^2}, \qquad \arg(z) = \operatorname{atan2}(b, a)$$

The modulus is the distance from the origin — the Euclidean norm of the pair, which is why §3.1’s norms carry over unchanged. The argument is the angle from the positive real axis.

In these terms, multiplication is simple:

$$\lvert z_1 z_2\rvert = \lvert z_1\rvert\,\lvert z_2\rvert, \qquad \arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$$

Moduli multiply, arguments add. Multiplying by $z$ scales by $\lvert z\rvert$ and rotates by $\arg(z)$. Check it on $i$: modulus $1$, argument $90°$. So multiplying by $i$ rotates by a quarter turn and changes no length — which is what we asked for at the start.

Euler’s formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

So any complex number factors into “how big” times “which way”:

$$z = \lvert z\rvert\, e^{i\arg(z)}$$

Two consequences used later:

• $e^{i\theta}$ always has modulus $1$ — it is a point on the unit circle, and $\theta$ is the angle.
• Multiplying two of them adds the exponents, which is the “arguments add” rule again, now as ordinary index arithmetic.

Setting $\theta = \pi$ gives $e^{i\pi} = -1$: a half turn is multiplication by $-1$, as promised.

Conjugates

$$\bar{z} = a - bi$$

Reflection across the real axis. Two properties do real work:

$$z\bar{z} = a^2 + b^2 = \lvert z\rvert^2, \qquad \overline{z_1 z_2} = \bar{z_1}\,\bar{z_2}$$

The first says a complex number times its conjugate is real and non-negative — which is how you divide by a complex number, and the reason $\lvert z\rvert^2$ appears wherever a magnitude is needed.

The second gives the fact §4.2 depends on:

Complex eigenvalues of a real matrix come in conjugate pairs

If $\mathbf{A}$ is real and $\mathbf{A}\mathbf{v} = \lambda\mathbf{v}$, take the conjugate of the whole equation. Conjugation leaves real entries alone and distributes over products, so

$$\mathbf{A}\bar{\mathbf{v}} = \bar{\lambda}\,\bar{\mathbf{v}}$$

meaning $\bar{\lambda}$ is an eigenvalue too, with eigenvector $\bar{\mathbf{v}}$. So complex eigenvalues of a real matrix always arrive in pairs $\lambda, \bar{\lambda}$ — never alone.

Immediate corollary: a real matrix of odd size must have at least one real eigenvalue, because the complex ones pair up and an odd count cannot be entirely made of pairs. A real $3\times 3$ rotation therefore always has a genuine axis. That is not a coincidence about three dimensions; it is this counting argument.

The one place you need this

The rotation matrix by angle $\theta$:

$$\mathbf{R}(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$$

Its characteristic polynomial is

$$\det(\mathbf{R} - \lambda\mathbf{I}) = (\cos\theta - \lambda)^2 + \sin^2\theta = \lambda^2 - 2\lambda\cos\theta + 1$$

with discriminant

$$4\cos^2\theta - 4 = -4\sin^2\theta \;\le\; 0$$

Negative whenever $\sin\theta \neq 0$. So for any rotation that is not by $0°$ or $180°$ the eigenvalues are complex:

$$\lambda = \cos\theta \pm i\sin\theta = e^{\pm i\theta}$$

Look at what that says. The eigenvalues of a rotation by $\theta$ are $e^{\pm i\theta}$: modulus $1$, argument $\pm\theta$. The matrix rotates by $\theta$; its eigenvalues are the rotation by $\theta$, written as complex numbers. Modulus $1$ says lengths are preserved, which is exactly §3.9’s claim that rotations are orthogonal transformations — recovered from the eigenvalues.

And “no real eigenvalue” now has a geometric meaning rather than an algebraic one: a rotation turns every direction, so there is no direction it merely scales.

Worked example by hand

Take $\theta = 90°$, so $\mathbf{R} = \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$.

Characteristic polynomial: $\lambda^2 - 2\lambda\cos 90° + 1 = \lambda^2 + 1$. Roots $\lambda = \pm i$.

Check the eigenvector for $\lambda = i$. We need $(\mathbf{R} - i\mathbf{I})\mathbf{v} = \mathbf{0}$:

$$\begin{bmatrix}-i & -1\\ 1 & -i\end{bmatrix}\begin{bmatrix}v_1\\ v_2\end{bmatrix} = \mathbf{0}$$

The second row gives $v_1 = i v_2$. Take $v_2 = 1$, so $\mathbf{v} = (i, 1)^\top$. Verify:

$$\mathbf{R}\mathbf{v} = \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}\begin{bmatrix}i\\ 1\end{bmatrix} = \begin{bmatrix}-1\\ i\end{bmatrix} = i\begin{bmatrix}i\\ 1\end{bmatrix} = i\mathbf{v} \quad\checkmark$$

using $i\cdot i = -1$ in the top entry. And the conjugate pair: $\bar{\lambda} = -i$ with $\bar{\mathbf{v}} = (-i, 1)^\top$.

Now arithmetic on $z = 3 + 4i$ and $w = 1 - 2i$, every quantity by hand:

quantity	working	result
$z + w$	$(3+1) + (4-2)i$	$4 + 2i$
$zw$	$(3\cdot1 - 4\cdot(-2)) + (3\cdot(-2) + 4\cdot1)i$	$11 - 2i$
$\lvert z\rvert$	$\sqrt{9 + 16}$	$5$
$\lvert w\rvert$	$\sqrt{1 + 4}$	$\sqrt5 \approx 2.2360680$
$\lvert zw\rvert$	$5\sqrt5$	$\approx 11.1803399$
$\arg(z)$	$\operatorname{atan2}(4, 3)$	$\approx 0.9272952$ rad, $53.13°$
$\arg(w)$	$\operatorname{atan2}(-2, 1)$	$\approx -1.1071487$ rad, $-63.43°$
$\arg(zw)$	$0.9272952 + (-1.1071487)$	$\approx -0.1798535$ rad, $-10.30°$
$z\bar{z}$	$(3+4i)(3-4i) = 9 + 16$	$25 = \lvert z\rvert^2$

Check the last modulus directly: $\lvert 11 - 2i\rvert = \sqrt{121 + 4} = \sqrt{125} = 5\sqrt5$. Matches. Moduli multiplied and arguments added, exactly as claimed.

See it move

Drag the two complex numbers and watch the product. The dashed circle has radius $\lvert z_1\rvert\lvert z_2\rvert$: the product always lands on it, because moduli multiply. The angles add head to tail.

sketch Multiplying complex numbers scales and rotates p5.js Pause Restart

Drag the modulus and argument of each factor. The product's modulus is the two moduli multiplied and its argument is the two arguments added, so it always sits on the dashed circle.

Set both moduli to $1$. The product stays on the unit circle no matter how you turn the arguments — pure rotation, no scaling. Those unit-modulus complex numbers are precisely the eigenvalues $e^{\pm i\theta}$ of $\mathbf{R}(\theta)$, and “modulus one” is “lengths preserved”.

From scratch

complex_numbers.py

import numpy as np
 
z, w = 3 + 4j, 1 - 2j          # Python writes the imaginary unit as j, not i
 
print("z + w :", z + w)                       # (4+2j)
print("z * w :", z * w)                       # (11-2j)
print("|z|   :", abs(z), " |w|:", round(abs(w), 7))
print("|zw| == |z||w| :", np.isclose(abs(z * w), abs(z) * abs(w)))
print("arg z :", round(np.angle(z), 7), " arg w:", round(np.angle(w), 7))
print("arg zw == arg z + arg w :",
      np.isclose(np.angle(z * w), np.angle(z) + np.angle(w)))
print("z * conj(z) :", z * z.conjugate(), " == |z|^2 =", abs(z) ** 2)
 
# Euler's formula, checked numerically.
theta = np.pi / 3
print("e^{i.theta} == cos + i sin :",
      np.isclose(np.exp(1j * theta), np.cos(theta) + 1j * np.sin(theta)))
print("e^{i.pi} :", np.round(np.exp(1j * np.pi), 12))
 
# The rotation matrix: real entries, complex eigenvalues.
theta = np.pi / 2
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])
vals, vecs = np.linalg.eig(R)
print("R is real:", np.isrealobj(R), " eigenvalues:", np.round(vals, 10))
print("moduli all 1:", np.allclose(np.abs(vals), 1.0))
print("conjugate pair:", np.isclose(vals[0], vals[1].conjugate()))
print("eigenvalues equal exp(+-i.theta):",
      np.allclose(np.sort_complex(vals),
                  np.sort_complex(np.array([np.exp(1j*theta), np.exp(-1j*theta)]))))
 
# Odd size forces a real eigenvalue: rotation about the z axis in 3-D.
R3 = np.array([[np.cos(theta), -np.sin(theta), 0.0],
               [np.sin(theta),  np.cos(theta), 0.0],
               [0.0,            0.0,           1.0]])
v3 = np.linalg.eigvals(R3)
print("3x3 rotation eigenvalues:", np.round(v3, 10))
print("has a real one (the axis):", np.any(np.isclose(v3.imag, 0.0)))

text

z + w : (4+2j)
z * w : (11-2j)
|z|   : 5.0  |w|: 2.236068
|zw| == |z||w| : True
arg z : 0.9272952  arg w: -1.1071487
arg zw == arg z + arg w : True
z * conj(z) : (25+0j)  == |z|^2 = 25.0
e^{i.theta} == cos + i sin : True
e^{i.pi} : (-1+0j)
R is real: True  eigenvalues: [0.+1.j 0.-1.j]
moduli all 1: True
conjugate pair: True
eigenvalues equal exp(+-i.theta): True
3x3 rotation eigenvalues: [0.+1.j 0.-1.j 1.+0.j]
has a real one (the axis): True

The last two lines are the counting argument, executed. The $2\times2$ rotation has no real eigenvalue; the $3\times3$ one has exactly one, and it is $1$ — the axis, which the rotation leaves completely alone.

Python writes j for the imaginary unit, not i

Python and NumPy use j for the imaginary unit, an electrical-engineering convention. 1j is the imaginary unit; bare j is an undefined name; 1i is a syntax error. And abs(z) on a complex number gives the modulus, not a componentwise absolute value.

Pitfalls

The square-root sign is ambiguous over the complex numbers

Every nonzero complex number has two square roots and no canonical “positive” one. Writing $i = \sqrt{-1}$ is a definition by convention, and manipulations like $\sqrt{a}\sqrt{b} = \sqrt{ab}$ can fail. Use $i^2 = -1$ as the defining property instead; it never misleads.

Complex eigenvalues are not an error in your code

np.linalg.eig returning a complex array for a real input matrix is correct and expected. Taking .real to “clean it up” throws away the rotation. If you need real output, the question to ask is whether your matrix is symmetric — symmetric real matrices are guaranteed real eigenvalues (§4.2’s spectral theorem), and np.linalg.eigh exploits that and returns real values.

Complex eigenvalues mean the eigenvectors are complex too

There is no real eigenvector to plot. A rotation genuinely has no invariant direction in the plane, so nothing was lost in translation — the geometry and the algebra agree.

Ordering does not exist

There is no meaningful $<$ on complex numbers, so “the largest eigenvalue” needs care. What is meant is largest modulus — the spectral radius. §4.2’s power iteration converges to the eigenvector of largest modulus, and that phrasing is deliberate.

Compare

matrix property	eigenvalues	practical consequence
real symmetric	all real, orthogonal eigenvectors	use eigh; the spectral theorem applies (§4.2)
real, not symmetric	may be complex, in conjugate pairs	use eig; expect a complex dtype
rotation $\mathbf{R}(\theta)$	$e^{\pm i\theta}$, modulus $1$	length preserving; no real invariant direction (§3.9)
real, odd size	at least one real eigenvalue	a 3-D rotation always has an axis
covariance matrix	real, all $\ge 0$	it is symmetric positive semidefinite (§6.4)

The first and last rows are the reason most of machine learning never meets a complex number: covariance matrices, Gram matrices and Hessians are all symmetric, and symmetric real matrices have real eigenvalues. Complex arithmetic shows up when a matrix is not symmetric — a rotation, a transition matrix, the Jacobian of a dynamical system.

pch.quizTag Check yourself

1. What happens to modulus and argument when two complex numbers are multiplied?

   A Both multiply B Both add C Moduli multiply and arguments add D Moduli add and arguments multiply
   pch.quizShowAnswer

   C — Moduli multiply and arguments add — Which is why multiplying by a complex number is a scale-and-rotate, and why a unit-modulus factor is a pure rotation.

2. Why can a real matrix have complex eigenvalues?

   A Because of floating-point error B Because the characteristic polynomial can have no real roots, even though its coefficients are real C Because the matrix was not square D It cannot — that would indicate a bug
   pch.quizShowAnswer

   B — Because the characteristic polynomial can have no real roots, even though its coefficients are real — A rotation's characteristic polynomial has discriminant minus four sine squared, which is negative for any genuine rotation. Real coefficients, complex roots, and the eigenvalues turn out to be e to the plus or minus i theta.

3. A real five-by-five matrix must have at least one real eigenvalue. Why?

   A All odd-sized matrices are symmetric B Complex eigenvalues of a real matrix come in conjugate pairs, and five cannot be made entirely of pairs C Because the determinant is real D It need not — five-by-five matrices can have all complex eigenvalues
   pch.quizShowAnswer

   B — Complex eigenvalues of a real matrix come in conjugate pairs, and five cannot be made entirely of pairs — Conjugating the eigenvalue equation shows the conjugate is also an eigenvalue, so complex ones arrive two at a time. An odd count leaves at least one unpaired, which therefore must be real — this is why a 3-D rotation always has an axis.

4. You call np.linalg.eig on a real covariance matrix and get a complex dtype with negligible imaginary parts. What is the right response?

   A Take the real part and move on B Use eigh instead, which exploits symmetry and returns real values C Add a small constant to the diagonal D Transpose the matrix first
   pch.quizShowAnswer

   B — Use eigh instead, which exploits symmetry and returns real values — A covariance matrix is symmetric, so the spectral theorem guarantees real eigenvalues; the imaginary parts are pure numerical noise from the general-purpose routine. eigh assumes symmetry, is faster, and returns real values by construction.

🧪 Try It Yourself

Exercise 1 – Multiplying by i is a quarter turn

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Exercise 2 – Moduli multiply

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Exercise 3 – Euler’s formula

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Exercise 4 – A rotation’s eigenvalues

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Exercise 5 – Symmetric matrices stay real

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From the book

Mathematics for Machine Learning keeps complex numbers to a minimum but does not avoid them. §4.2 defines eigenvalues as roots of the characteristic polynomial (Definition 4.6, Equation 4.22) and remarks explicitly that eigenvalues may be complex even for a real matrix. Example 4.7 works a matrix whose eigenvalues are complex, and §3.9’s rotation matrix (Equation 3.76) is the case where this is unavoidable. Theorem 4.15, the spectral theorem, is the counterweight: a symmetric matrix has real eigenvalues and an orthonormal eigenbasis, which is why the covariance matrices of §6.4 and the PCA of Chapter 10 stay entirely real. $\mathbb{C}$ appears in the Table of Symbols on page 6.

Recall card

• Read $i$ as the quarter turn, not as the square root of minus one — the geometry then explains the algebra.
• A complex number is a point in the plane with one extra structure real pairs lack: a multiplication.
• Multiplying multiplies moduli and adds arguments, so multiplication is a scale-and-rotate and a unit-modulus factor is a pure rotation.
• Euler’s formula — e to the i theta is the unit-circle point at angle theta, so any complex number factors into size times direction.
• A number times its conjugate is its modulus squared, real and non-negative.
• Complex eigenvalues of a real matrix come in conjugate pairs, because conjugating the eigenvalue equation leaves a real matrix alone.
• An odd-sized real matrix therefore has at least one real eigenvalue — which is why a 3-D rotation always has an axis.
• A rotation by theta has eigenvalues e to the plus or minus i theta — modulus one, meaning lengths are preserved, with no real invariant direction.
• Symmetric real matrices always have real eigenvalues, which is why covariance matrices, Gram matrices and Hessians never need complex arithmetic. Use eigh, not eig.
• There is no ordering on the complex numbers, so “largest eigenvalue” always means largest modulus.

Next: the tool you will check every derivation against — NumPy for Mathematics.