Chapter 127. Complex Vector Spaces

127.1 Introduction

A complex vector space is a vector space whose scalars are complex numbers.

In real linear algebra, vectors are multiplied by real scalars from the field $\mathbb{R}$ . In complex linear algebra, vectors are multiplied by complex scalars from the field $\mathbb{C}$ .

The transition from real to complex scalars changes many structural properties of linear algebra. Polynomials factor more completely. Eigenvalues always exist for square matrices. Inner products require conjugation. Orthogonality becomes Hermitian rather than symmetric.

Complex vector spaces form the natural setting for large areas of mathematics and physics, including Fourier analysis, quantum mechanics, signal processing, representation theory, and spectral theory.

A large part of advanced linear algebra is simpler over $\mathbb{C}$ than over $\mathbb{R}$ .

127.2 The Complex Numbers

The complex numbers form the field

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

where

i^2 = -1.

The number $a$ is called the real part:

\operatorname{Re}(z) = a,

and $b$ is called the imaginary part:

\operatorname{Im}(z) = b.

Addition and multiplication are defined by

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

and

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

The complex conjugate of

z = a+bi

\overline{z} = a-bi.

Complex conjugation satisfies

\overline{z+w} = \overline{z} + \overline{w},

\overline{zw} = \overline{z}\,\overline{w},

and

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

The modulus of $z$ is

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

Complex conjugation plays a central role in complex linear algebra.

127.3 Definition of a Complex Vector Space

A complex vector space is a set $V$ equipped with:

Vector addition

+: V \times V \to V,

Scalar multiplication

\mathbb{C} \times V \to V,

satisfying the vector space axioms.

These axioms include:

Property	Formula
Associativity of addition	$u+(v+w)=(u+v)+w$
Commutativity of addition	$u+v=v+u$
Additive identity	$v+0=v$
Additive inverse	$v+(-v)=0$
Scalar distributivity	$a(u+v)=au+av$
Vector distributivity	$(a+b)v=av+bv$
Associativity of scalars	$a(bv)=(ab)v$
Scalar identity	$1v=v$

The only difference from real vector spaces is the field of scalars.

127.4 Examples

Example 1. Complex Coordinate Space

The set

\mathbb{C}^n

is a complex vector space.

Vectors have the form

\begin{bmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \end{bmatrix}, \qquad z_i \in \mathbb{C}.

Addition and scalar multiplication are componentwise.

Example 2. Complex Matrices

The set of all $m \times n$ complex matrices,

M_{m,n}(\mathbb{C}),

forms a complex vector space.

Example 3. Complex Polynomials

The set

\mathbb{C}[x]

of polynomials with complex coefficients forms a complex vector space.

Example 4. Complex-Valued Functions

The set of functions

f : X \to \mathbb{C}

forms a complex vector space under pointwise operations.

Example 5. Fourier Modes

Functions of the form

e^{inx}, \qquad n \in \mathbb{Z},

are naturally complex-valued and generate spaces used in Fourier analysis.

127.5 Real and Complex Dimensions

A complex vector space may also be viewed as a real vector space by restricting scalars from $\mathbb{C}$ to $\mathbb{R}$ .

This changes the dimension.

For example,

\mathbb{C}^n

has:

Scalar field	Dimension
Over $\mathbb{C}$	$n$
Over $\mathbb{R}$	$2n$

The reason is that each complex coordinate contains two real parameters.

For example,

a+bi

corresponds to

(a,b).

Thus

\mathbb{C}

is one-dimensional over $\mathbb{C}$ but two-dimensional over $\mathbb{R}$ .

The scalar field must always be specified when discussing dimension.

127.6 Linear Combinations

In a complex vector space, linear combinations use complex coefficients.

v_1,\ldots,v_n \in V

and

c_1,\ldots,c_n \in \mathbb{C},

then

c_1v_1 + \cdots + c_nv_n

is a complex linear combination.

For example, in $\mathbb{C}^2$ ,

(1+i) \begin{bmatrix} 1 \\ 0 \end{bmatrix} + (2-3i) \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1+i \\ 2-3i \end{bmatrix}.

All concepts such as span, basis, and linear independence are defined exactly as in real vector spaces, except that scalars are complex.

127.7 Bases and Dimension

A basis of a complex vector space is a linearly independent spanning set.

\mathcal{B} = \{v_1,\ldots,v_n\}

is a basis, then every vector has a unique representation

v = c_1v_1 + \cdots + c_nv_n, \qquad c_i \in \mathbb{C}.

The number of basis vectors is the dimension of the space.

For example, the standard basis of $\mathbb{C}^n$ is

e_1,\ldots,e_n,

where

e_i = \begin{bmatrix} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{bmatrix}.

All finite-dimensional complex vector spaces of dimension $n$ are isomorphic to $\mathbb{C}^n$ .

127.8 Complex Matrices

A complex matrix has complex entries:

A = (a_{ij}), \qquad a_{ij} \in \mathbb{C}.

Matrix operations are unchanged algebraically.

For example,

A+B, \qquad AB, \qquad cA,

are defined exactly as before.

The transpose of a matrix is

A^T.

The complex conjugate matrix is

\overline{A}.

The conjugate transpose is

A^* = \overline{A}^T.

This operation replaces the transpose in many important theorems.

127.9 Hermitian Inner Products

Inner products over complex vector spaces require conjugation.

A Hermitian inner product on $V$ is a function

\langle \cdot,\cdot \rangle : V \times V \to \mathbb{C}

satisfying:

Property	Formula
Conjugate symmetry	$\langle u,v\rangle = \overline{\langle v,u\rangle}$
Linearity in first variable	$\langle au+bw,v\rangle = a\langle u,v\rangle + b\langle w,v\rangle$
Positive definiteness	$\langle v,v\rangle > 0$ for $v\neq0$

The standard Hermitian inner product on $\mathbb{C}^n$ is

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

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

Without conjugation, positive definiteness fails.

For example,

i^2 = -1.

Thus

\langle i,i\rangle

would become negative without conjugation.

127.10 Norms

The norm induced by the Hermitian inner product is

\|v\| = \sqrt{\langle v,v\rangle}.

For

v = \begin{bmatrix} z_1 \\ \vdots \\ z_n \end{bmatrix},

the standard norm is

\|v\| = \sqrt{ |z_1|^2+\cdots+|z_n|^2 }.

This extends Euclidean geometry to complex spaces.

127.11 Orthogonality

Vectors $u$ and $v$ are orthogonal if

\langle u,v\rangle = 0.

Orthogonality behaves similarly to the real case, but conjugation affects computations.

For example, if

u = \begin{bmatrix} 1 \\ i \end{bmatrix}, \qquad v = \begin{bmatrix} 1 \\ -i \end{bmatrix},

then

\langle u,v\rangle = 1\cdot1 + i\cdot\overline{-i} = 1+i(i) = 1-1 = 0.

Thus the vectors are orthogonal.

127.12 Unitary Matrices

A complex matrix $U$ is unitary if

U^*U = I.

U^*U = I

This generalizes orthogonal matrices.

Equivalent conditions include:

Condition	Meaning
$U^*U=I$	Inverse equals conjugate transpose
$U^{-1}=U^*$	Length preservation
Columns orthonormal	Geometric interpretation

Unitary matrices preserve inner products:

\langle Ux,Uy\rangle = \langle x,y\rangle.

They also preserve norms and orthogonality.

Examples include complex rotations and Fourier transform matrices.

127.13 Hermitian Matrices

A matrix $A$ is Hermitian if

A^* = A.

A^* = A

Hermitian matrices generalize symmetric matrices.

Their entries satisfy

a_{ij} = \overline{a_{ji}}.

Hermitian matrices have several important properties:

Property	Statement
Eigenvalues	Real
Eigenvectors for distinct eigenvalues	Orthogonal
Diagonalization	Unitary diagonalization exists

These properties are fundamental in quantum mechanics and spectral theory.

127.14 The Spectral Theorem

The spectral theorem for complex spaces states:

If $A$ is Hermitian, then there exists a unitary matrix $U$ such that

A = UDU^*,

where $D$ is diagonal with real entries.

A = UDU^*

This theorem says that Hermitian matrices admit orthonormal eigenbases.

More generally, every normal matrix satisfies a unitary diagonalization theorem.

A matrix is normal if

AA^* = A^*A.

The class of normal matrices includes:

Type	Condition
Hermitian	$A^*=A$
Skew-Hermitian	$A^*=-A$
Unitary	$A^*A=I$

127.15 Complex Eigenvalues

Over $\mathbb{C}$ , every polynomial factors completely.

This yields one of the major advantages of complex linear algebra:

Every square complex matrix has at least one eigenvalue.

This follows from the Fundamental Theorem of Algebra.

p_A(\lambda) = \det(A-\lambda I),

then $p_A$ factors into linear terms over $\mathbb{C}$ :

p_A(\lambda) = (\lambda-\lambda_1)\cdots(\lambda-\lambda_n).

Thus complex matrices always possess eigenvalues.

This property fails over $\mathbb{R}$ .

For example,

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

has no real eigenvalues but has complex eigenvalues

\pm i.

127.16 Complexification

Given a real vector space $V$ , one may construct its complexification:

V_{\mathbb{C}} = V \otimes_{\mathbb{R}} \mathbb{C}.

Intuitively, this allows complex scalar multiplication.

If $V=\mathbb{R}^n$ , then

V_{\mathbb{C}} \cong \mathbb{C}^n.

Complexification allows real operators to be studied using complex spectral methods.

This is standard in differential equations and functional analysis.

127.17 Applications

Complex vector spaces appear throughout mathematics and science.

Quantum Mechanics

Quantum states are vectors in complex Hilbert spaces.

Observables correspond to Hermitian operators.

Unitary operators describe time evolution.

Fourier Analysis

Complex exponentials

e^{inx}

form orthogonal bases for periodic functions.

Signal Processing

Complex numbers encode amplitude and phase simultaneously.

Differential Equations

Complex eigenvalues describe oscillatory behavior.

Representation Theory

Representations are often studied over $\mathbb{C}$ because decomposition theory becomes simpler.

127.18 Infinite-Dimensional Complex Spaces

Complex vector spaces need not be finite-dimensional.

Examples include:

Space	Description
$L^2(\mathbb{R})$	Square-integrable complex functions
$\ell^2$	Square-summable sequences
Hardy spaces	Holomorphic function spaces
Hilbert spaces	Complete inner product spaces

These spaces form the foundation of functional analysis.

127.19 Summary

A complex vector space is a vector space over the field $\mathbb{C}$ . Most constructions from real linear algebra extend naturally, but complex conjugation changes the structure of inner products, orthogonality, adjoints, and spectral theory.

Complex vector spaces have several important advantages:

Feature	Consequence
Algebraic closure of $\mathbb{C}$	Eigenvalues always exist
Hermitian inner products	Stable geometric structure
Unitary operators	Norm-preserving transformations
Spectral theorem	Orthonormal diagonalization
Complex exponentials	Fourier analysis and oscillations

Complex linear algebra provides the natural framework for modern spectral theory, quantum mechanics, harmonic analysis, and advanced operator theory.