Appendix C. Real and Complex Numbers

Linear algebra is usually developed over a field of scalars. The two most common scalar fields are the real numbers $\mathbb{R}$ and the complex numbers $\mathbb{C}$ . A real vector space allows real scalar multiplication. A complex vector space allows complex scalar multiplication. This distinction affects eigenvalues, inner products, matrix factorizations, and spectral theory.

C.1 The Real Numbers

The real numbers form the number system used for ordinary measurement. They include integers, rational numbers, and irrational numbers.

\mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}.

A real number may be positive, negative, or zero. It can be placed on the real line. Addition and multiplication of real numbers satisfy the usual algebraic laws:

Law	Formula
Associativity of addition	$(a+b)+c=a+(b+c)$
Commutativity of addition	$a+b=b+a$
Additive identity	$a+0=a$
Additive inverse	$a+(-a)=0$
Associativity of multiplication	$(ab)c=a(bc)$
Commutativity of multiplication	$ab=ba$
Multiplicative identity	$a\cdot 1=a$
Multiplicative inverse	$a\neq 0 \implies a^{-1}\text{ exists}$
Distributivity	$a(b+c)=ab+ac$

These laws make $\mathbb{R}$ a field. A field is a scalar system in which addition, subtraction, multiplication, and division by nonzero elements are possible.

C.2 Order on the Real Numbers

The real numbers are ordered. For real numbers $a$ and $b$ , one may write

a < b, \qquad a \leq b, \qquad a > b, \qquad a \geq b.

The order is compatible with addition:

a \leq b \implies a+c \leq b+c.

It is also compatible with multiplication by nonnegative numbers:

a \leq b,\ c \geq 0 \implies ac \leq bc.

If $c < 0$ , the inequality reverses:

a \leq b,\ c < 0 \implies ac \geq bc.

Order is important in topics such as length, norm, positivity, optimization, and positive definite matrices. Complex numbers do not have an order compatible with field operations in the same way.

C.3 Absolute Value

The absolute value of a real number $x$ is

|x| = \begin{cases} x, & x \geq 0, \\ -x, & x < 0. \end{cases}

It measures distance from zero on the real line.

For example,

|5|=5, \qquad |-5|=5.

The absolute value satisfies:

Property	Formula
Nonnegativity	(
Definiteness	(
Multiplicativity	(
Triangle inequality	(

The triangle inequality is the one-dimensional prototype of norm inequalities in vector spaces.

C.4 Distance on the Real Line

The distance between two real numbers $a$ and $b$ is

|a-b|.

For example, the distance between $3$ and $-2$ is

|3-(-2)| = |5| = 5.

This formula generalizes to Euclidean distance in $\mathbb{R}^n$ . If

x=(x_1,\ldots,x_n), \qquad y=(y_1,\ldots,y_n),

then the Euclidean distance is

\sqrt{(x_1-y_1)^2+\cdots+(x_n-y_n)^2}.

Thus the absolute value is the first example of a norm.

C.5 Square Roots and Positivity

For every nonnegative real number $a$ , there exists a unique nonnegative real number $\sqrt{a}$ such that

(\sqrt{a})^2=a.

For example,

\sqrt{9}=3.

The equation

x^2=a

has two real solutions when $a>0$ :

x=\sqrt{a} \quad \text{and} \quad x=-\sqrt{a}.

It has one real solution when $a=0$ , and no real solution when $a<0$ .

The failure of the equation

x^2=-1

to have a real solution leads to the complex numbers.

C.6 The Complex Numbers

The complex numbers extend the real numbers by adjoining a new number $i$ , called the imaginary unit, satisfying

i^2=-1.

Every complex number has the form

z=a+bi,

where $a,b\in\mathbb{R}$ . The number $a$ is the real part of $z$ , and $b$ is the imaginary part of $z$ . The set of all complex numbers is denoted by $\mathbb{C}$ .

We write

\operatorname{Re}(z)=a, \qquad \operatorname{Im}(z)=b.

For example, if

z=3-4i,

then

\operatorname{Re}(z)=3, \qquad \operatorname{Im}(z)=-4.

A real number is a complex number with imaginary part zero:

a = a+0i.

Thus

\mathbb{R} \subseteq \mathbb{C}.

C.7 Addition and Multiplication in $\mathbb{C}$

Complex numbers are added componentwise:

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

For example,

(2+3i)+(5-i)=7+2i.

Multiplication is defined by the distributive law and the relation $i^2=-1$ :

(a+bi)(c+di) = ac+adi+bci+bd i^2.

Since $i^2=-1$ ,

(a+bi)(c+di) = (ac-bd)+(ad+bc)i.

For example,

(2+3i)(4-i) = 8-2i+12i-3i^2 = 11+10i.

The complex numbers form a field. Addition, subtraction, multiplication, and division by nonzero complex numbers are all defined.

C.8 Complex Conjugation

The complex conjugate of

z=a+bi

\overline{z}=a-bi.

Conjugation changes the sign of the imaginary part and leaves the real part unchanged.

For example,

\overline{3+5i}=3-5i.

Important identities include:

Identity	Formula
Conjugate of a sum	$\overline{z+w}=\overline{z}+\overline{w}$
Conjugate of a product	$\overline{zw}=\overline{z}\,\overline{w}$
Double conjugation	$\overline{\overline{z}}=z$
Real criterion	$z\in\mathbb{R}\iff z=\overline{z}$

Conjugation is essential in complex inner product spaces. It appears in Hermitian matrices, unitary matrices, and the complex spectral theorem.

C.9 Modulus of a Complex Number

The modulus of

z=a+bi

|z|=\sqrt{a^2+b^2}.

Geometrically, this is the distance from the origin to the point $(a,b)$ in the complex plane.

The modulus satisfies

|z| \geq 0,

|z|=0 \iff z=0,

|zw|=|z||w|,

and

|z+w|\leq |z|+|w|.

The product of a complex number with its conjugate is real and nonnegative:

z\overline{z}=|z|^2.

Indeed, if $z=a+bi$ , then

z\overline{z} = (a+bi)(a-bi) = a^2+b^2.

This identity is used to divide by complex numbers.

C.10 Division in $\mathbb{C}$

If $z\neq 0$ , then

z^{-1}=\frac{\overline{z}}{|z|^2}.

For

z=a+bi,

this gives

\frac{1}{a+bi} = \frac{a-bi}{a^2+b^2}.

For example,

\frac{1}{2+3i} = \frac{2-3i}{2^2+3^2} = \frac{2}{13}-\frac{3}{13}i.

Thus division by a nonzero complex number reduces to multiplication by its conjugate divided by its squared modulus.

C.11 The Complex Plane

A complex number

z=a+bi

can be represented by the point $(a,b)$ in the plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

This identifies $\mathbb{C}$ with $\mathbb{R}^2$ as a real vector space:

a+bi \leftrightarrow (a,b).

However, $\mathbb{C}$ has more structure than $\mathbb{R}^2$ , because complex numbers can be multiplied.

Multiplication by $i$ sends

a+bi

i(a+bi)=ai+b i^2=-b+ai.

In coordinate form,

(a,b) \mapsto (-b,a).

This is rotation by $90^\circ$ counterclockwise.

Thus complex multiplication has a geometric interpretation: it combines scaling and rotation.

C.12 Polar Form

A nonzero complex number can be written in polar form:

z=r(\cos\theta+i\sin\theta),

where

r=|z|

and $\theta$ is an argument of $z$ .

The number $r$ is the distance from the origin. The angle $\theta$ is measured from the positive real axis.

z=a+bi,

then

a=r\cos\theta, \qquad b=r\sin\theta.

Polar form is useful because multiplication becomes simple:

r(\cos\theta+i\sin\theta)\,s(\cos\phi+i\sin\phi) = rs(\cos(\theta+\phi)+i\sin(\theta+\phi)).

Thus multiplying complex numbers multiplies their moduli and adds their arguments.

C.13 Euler’s Formula

Euler’s formula states that

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

Using this notation, polar form becomes

z=re^{i\theta}.

Multiplication is then

(re^{i\theta})(se^{i\phi})=rs e^{i(\theta+\phi)}.

Euler’s formula connects algebra, geometry, and analysis. In linear algebra, it appears in rotations, complex eigenvalues, Fourier analysis, unitary matrices, and matrix exponentials.

C.14 Real and Complex Vector Spaces

A real vector space uses scalars from $\mathbb{R}$ . A complex vector space uses scalars from $\mathbb{C}$ .

For example, $\mathbb{R}^n$ is a real vector space. Its vectors have real components, and scalar multiplication uses real numbers.

The space $\mathbb{C}^n$ is a complex vector space. Its vectors have complex components, and scalar multiplication uses complex numbers.

Every complex vector space can also be viewed as a real vector space by restricting scalars from $\mathbb{C}$ to $\mathbb{R}$ . For example, $\mathbb{C}$ has dimension $1$ over $\mathbb{C}$ , but dimension $2$ over $\mathbb{R}$ .

Indeed,

\mathbb{C} = \{a+bi : a,b\in\mathbb{R}\}.

As a real vector space, a basis is

\{1,i\}.

As a complex vector space, a basis is

\{1\}.

The scalar field affects dimension.

C.15 Real Matrices and Complex Matrices

A real matrix has entries in $\mathbb{R}$ . A complex matrix has entries in $\mathbb{C}$ .

A real matrix may still have complex eigenvalues. For example,

A= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

represents rotation by $90^\circ$ in $\mathbb{R}^2$ . Its characteristic polynomial is

\lambda^2+1.

This polynomial has no real roots. Over $\mathbb{C}$ , it has roots

\lambda=i \quad \text{and} \quad \lambda=-i.

Thus complex numbers are needed for a complete eigenvalue theory. The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has a complex root, so polynomial equations behave more completely over $\mathbb{C}$ than over $\mathbb{R}$ .

C.16 Conjugate Transpose

For a complex matrix $A$ , the transpose alone is usually not the correct analogue of the real transpose. Instead one uses the conjugate transpose.

A=(a_{ij}),

then the conjugate transpose of $A$ is denoted by

A^*

and is defined by

(A^*)_{ij}=\overline{a_{ji}}.

Thus $A^*$ is obtained by transposing $A$ and conjugating each entry.

For example, if

A= \begin{bmatrix} 1+i & 2 \\ 3i & 4-i \end{bmatrix},

then

A^* = \begin{bmatrix} 1-i & -3i \\ 2 & 4+i \end{bmatrix}.

The conjugate transpose is central in complex inner product spaces.

C.17 Real and Complex Inner Products

On $\mathbb{R}^n$ , the standard inner product is

x \cdot y = x_1y_1+\cdots+x_ny_n.

On $\mathbb{C}^n$ , the standard inner product usually includes complex conjugation:

\langle x,y\rangle = \overline{x_1}y_1+\cdots+\overline{x_n}y_n.

Some authors place the conjugate on the second variable instead. This book uses the convention above unless otherwise stated.

The conjugate is needed so that

\langle x,x\rangle

is real and nonnegative.

Indeed, if

x=(x_1,\ldots,x_n),

then

\langle x,x\rangle = |x_1|^2+\cdots+|x_n|^2.

This quantity is zero only when $x=0$ .

C.18 Hermitian and Unitary Matrices

A complex square matrix $A$ is Hermitian if

A^*=A.

Hermitian matrices are the complex analogue of real symmetric matrices. Their eigenvalues are real, and they have strong orthogonality properties.

A complex square matrix $U$ is unitary if

U^*U=UU^*=I.

Unitary matrices are the complex analogue of real orthogonal matrices. They preserve inner products and norms.

In real linear algebra, the corresponding conditions are

A^T=A

for symmetric matrices and

Q^TQ=QQ^T=I

for orthogonal matrices.

C.19 Choosing the Scalar Field

The choice between $\mathbb{R}$ and $\mathbb{C}$ depends on the problem.

Use $\mathbb{R}$ when	Use $\mathbb{C}$ when
Quantities are naturally real-valued	Eigenvalues may be complex
Geometry takes place in real space	Rotations and oscillations are central
Order and positivity matter directly	Polynomial factorization is important
Optimization uses real variables	Fourier methods are used
Symmetric matrices are enough	Hermitian and unitary structure appears

Many real problems are temporarily extended to $\mathbb{C}$ because the complex setting gives cleaner algebra. After solving the complex problem, one may return to the real interpretation.

C.20 Summary

The real numbers provide order, distance, positivity, and the usual scalar system for geometry and computation. The complex numbers extend the real numbers by introducing $i$ , where $i^2=-1$ . Every complex number has the form $a+bi$ , and complex arithmetic follows from ordinary algebra together with this defining relation.

For linear algebra, the main points are:

Concept	Real case	Complex case
Scalar field	$\mathbb{R}$	$\mathbb{C}$
Standard space	$\mathbb{R}^n$	$\mathbb{C}^n$
Transpose analogue	$A^T$	$A^*$
Symmetric analogue	Symmetric matrix	Hermitian matrix
Orthogonal analogue	Orthogonal matrix	Unitary matrix
Eigenvalue theory	May require complex roots	Algebraically complete

Real and complex numbers are both fundamental. Real numbers support geometry and measurement. Complex numbers complete the algebraic picture and make spectral theory more natural.

Appendix C. Real and Complex Numbers

C.1 The Real Numbers

C.2 Order on the Real Numbers

C.3 Absolute Value

C.4 Distance on the Real Line

C.5 Square Roots and Positivity

C.6 The Complex Numbers

C.7 Addition and Multiplication in C\mathbb{C}C

C.8 Complex Conjugation

C.9 Modulus of a Complex Number

C.10 Division in C\mathbb{C}C

C.11 The Complex Plane

C.12 Polar Form

C.13 Euler’s Formula

C.14 Real and Complex Vector Spaces

C.15 Real Matrices and Complex Matrices

C.16 Conjugate Transpose

C.17 Real and Complex Inner Products

C.18 Hermitian and Unitary Matrices

C.19 Choosing the Scalar Field

C.20 Summary

C.7 Addition and Multiplication in $\mathbb{C}$

C.10 Division in $\mathbb{C}$