Chapter 55. Hermitian Spaces

A Hermitian space is a complex vector space equipped with a Hermitian inner product. It is the complex analogue of a real inner product space. The difference is conjugation. In complex vector spaces, symmetry is replaced by conjugate symmetry, and transposes are replaced by conjugate transposes.

The standard model is $\mathbb{C}^n$ , where the inner product is

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

Some texts use the opposite convention, linear in the first argument instead of the second. The essential structure is the same, provided the convention is used consistently. A complex inner product is conjugate symmetric, linear in one argument, conjugate-linear in the other, and positive definite.

55.1 Complex Vector Spaces

A complex vector space is a vector space whose scalars are complex numbers. If $V$ is a complex vector space, then scalar multiplication allows

(\alpha,v)\mapsto \alpha v

for every

\alpha\in\mathbb{C}, \qquad v\in V.

The presence of complex scalars changes the inner product structure. In a real vector space, an inner product may be symmetric:

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

In a complex vector space, this condition must be modified. The correct condition is

\langle x,y\rangle = \overline{\langle y,x\rangle}.

This is conjugate symmetry.

Without conjugation, the quantity $\langle x,x\rangle$ need not be real or nonnegative. For example, if one tried to define

\langle z,w\rangle=zw

on $\mathbb{C}$ , then

\langle i,i\rangle=i^2=-1.

This cannot represent a squared length. The standard complex inner product instead gives

\langle i,i\rangle=\overline{i}i=(-i)i=1.

55.2 Definition of a Hermitian Inner Product

Let $V$ be a complex vector space. A Hermitian inner product on $V$ is a function

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

satisfying the following properties for all $u,v,w\in V$ and all $\alpha,\beta\in\mathbb{C}$ .

First, linearity in the second argument:

\langle u,\alpha v+\beta w\rangle = \alpha\langle u,v\rangle+\beta\langle u,w\rangle.

Second, conjugate linearity in the first argument:

\langle \alpha u+\beta v,w\rangle = \overline{\alpha}\langle u,w\rangle + \overline{\beta}\langle v,w\rangle.

Third, conjugate symmetry:

\langle u,v\rangle = \overline{\langle v,u\rangle}.

Fourth, positive definiteness:

\langle v,v\rangle>0 \quad \text{for every }v\ne 0.

A complex vector space equipped with such an inner product is called a Hermitian space.

Some mathematics texts place linearity in the first argument instead. Physics texts often use linearity in the second argument. Both conventions define the same kind of geometry, but formulas involving matrices and adjoints must match the chosen convention.

55.3 The Standard Hermitian Product on $\mathbb{C}^n$

For vectors

x= \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix}, \qquad y= \begin{bmatrix} y_1\\ y_2\\ \vdots\\ y_n \end{bmatrix}

in $\mathbb{C}^n$ , the standard Hermitian inner product is

\langle x,y\rangle = x^*y = \overline{x_1}y_1+\overline{x_2}y_2+\cdots+\overline{x_n}y_n.

Here

x^*=\overline{x}^{\,T}

is the conjugate transpose of $x$ .

For example, let

x= \begin{bmatrix} 1+i\\ 2 \end{bmatrix}, \qquad y= \begin{bmatrix} 3\\ i \end{bmatrix}.

Then

\langle x,y\rangle = \overline{1+i}\cdot 3+\overline{2}\cdot i.

Since

\overline{1+i}=1-i,

we get

\langle x,y\rangle = (1-i)3+2i = 3-3i+2i = 3-i.

Reversing the order gives

\langle y,x\rangle = \overline{3}(1+i)+\overline{i}\cdot 2 = 3(1+i)-2i = 3+i.

Thus

\langle x,y\rangle = \overline{\langle y,x\rangle}.

55.4 Norm in a Hermitian Space

A Hermitian inner product defines a norm by

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

In $\mathbb{C}^n$ ,

\|x\|_2 = \sqrt{|x_1|^2+|x_2|^2+\cdots+|x_n|^2}.

For

x= \begin{bmatrix} 1+i\\ 2 \end{bmatrix},

we have

\|x\|_2^2 = |1+i|^2+|2|^2.

Since

|1+i|^2=2, \qquad |2|^2=4,

we get

\|x\|_2^2=6,

and hence

\|x\|_2=\sqrt{6}.

The squared norm is always real and nonnegative. This follows from conjugate symmetry:

\langle v,v\rangle = \overline{\langle v,v\rangle},

so $\langle v,v\rangle$ is real, and positive definiteness makes it positive for $v\ne 0$ .

55.5 Orthogonality

Two vectors $u,v$ in a Hermitian space are orthogonal if

\langle u,v\rangle=0.

Because of conjugate symmetry,

\langle u,v\rangle=0

if and only if

\langle v,u\rangle=0.

Thus orthogonality remains symmetric.

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

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

Then

\langle u,v\rangle = \overline{1}\cdot 1+\overline{i}\cdot i = 1+(-i)i = 2.

So $u$ and $v$ are not orthogonal.

Now let

w= \begin{bmatrix} 1\\ -i \end{bmatrix}.

Then

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

Thus $u\perp w$ .

55.6 Orthonormal Sets

A list

q_1,\ldots,q_k

in a Hermitian space is orthonormal if

\langle q_i,q_j\rangle=\delta_{ij}.

That is,

\langle q_i,q_i\rangle=1

and

\langle q_i,q_j\rangle=0 \quad \text{when }i\ne j.

Every orthonormal set is linearly independent. Suppose

c_1q_1+\cdots+c_kq_k=0.

Take the inner product with $q_j$ . Using linearity in the second argument and conjugate linearity in the first, one obtains

c_j=0

under the convention used here by taking

\langle q_j,c_1q_1+\cdots+c_kq_k\rangle=0.

Since this holds for every $j$ , all coefficients vanish. Hence the set is linearly independent.

55.7 Coordinates in an Orthonormal Basis

Let

q_1,\ldots,q_n

be an orthonormal basis for a Hermitian space $V$ , using the convention that the inner product is linear in the second argument.

Every vector $v\in V$ has the expansion

v = \sum_{j=1}^n q_j\langle q_j,v\rangle.

The coefficient of $q_j$ is

\langle q_j,v\rangle.

This differs in appearance from the formula used under the opposite convention. With linearity in the first argument, the coefficient is usually written $\langle v,q_j\rangle$ . The geometry is unchanged, but the placement of the vector inside the inner product matters.

In matrix form, if

Q= \begin{bmatrix} |&|&&|\\ q_1&q_2&\cdots&q_n\\ |&|&&| \end{bmatrix},

then

Q^*Q=I,

and the coordinate vector of $v$ in this orthonormal basis is

Q^*v.

55.8 Hermitian Matrices

A complex square matrix $A$ is Hermitian if

A^*=A.

Equivalently,

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

for every $i,j$ .

For example,

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

is Hermitian, because the off-diagonal entries are complex conjugates and the diagonal entries are real.

Every Hermitian matrix has real diagonal entries. Indeed, if $A=A^*$ , then

a_{ii}=\overline{a_{ii}},

so $a_{ii}\in\mathbb{R}$ .

Hermitian matrices are the complex analogue of real symmetric matrices.

55.9 Hermitian Forms

A Hermitian form on $\mathbb{C}^n$ often has the form

\langle x,y\rangle_A=x^*Ay,

where $A$ is Hermitian.

For this to be an inner product, $A$ must also be positive definite:

x^*Ax>0 \quad \text{for every }x\ne 0.

If $A$ is Hermitian positive definite, then

\langle x,y\rangle_A=x^*Ay

is a Hermitian inner product.

The matrix $A$ changes the geometry of the space. It can weight different directions differently and make some directions longer than others.

55.10 Positive Definite Hermitian Matrices

A Hermitian matrix $A$ is positive definite if

x^*Ax>0 \quad \text{for every nonzero }x\in\mathbb{C}^n.

It is positive semidefinite if

x^*Ax\ge 0 \quad \text{for every }x\in\mathbb{C}^n.

Positive definite Hermitian matrices define inner products. Positive semidefinite Hermitian matrices define seminorms, where a nonzero vector may have length zero.

For example,

A= \begin{bmatrix} 2&0\\ 0&1 \end{bmatrix}

is Hermitian positive definite, and

\langle x,y\rangle_A=2\overline{x_1}y_1+\overline{x_2}y_2.

The first coordinate contributes twice as much as the second coordinate to the squared norm.

55.11 The Adjoint of a Linear Map

Let $T:V\to V$ be a linear map on a finite-dimensional Hermitian space. The adjoint of $T$ is the unique linear map $T^*$ satisfying

\langle T^*u,v\rangle=\langle u,Tv\rangle

for all $u,v\in V$ , using the convention that the inner product is linear in the second argument.

In $\mathbb{C}^n$ with the standard Hermitian product, the adjoint is represented by the conjugate transpose matrix.

If $T$ is represented by $A$ , then $T^*$ is represented by

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

The adjoint is the Hermitian-space analogue of the transpose in real Euclidean space.

55.12 Self-Adjoint Operators

A linear operator $T$ on a Hermitian space is self-adjoint if

T^*=T.

In matrix form, this means

A^*=A.

Thus self-adjoint operators are represented by Hermitian matrices.

Self-adjoint operators are important because they have real eigenvalues and orthogonal eigenspaces. They are the complex analogue of real symmetric operators.

Av=\lambda v

and $A=A^*$ , then

\lambda = \frac{v^*Av}{v^*v}.

Since $v^*Av$ is real for Hermitian $A$ , $\lambda$ must be real.

55.13 Unitary Operators

A linear operator $U$ on a Hermitian space is unitary if

U^*U=I.

Equivalently,

U^{-1}=U^*.

Unitary operators preserve the Hermitian inner product:

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

They also preserve norm:

\|Ux\|=\|x\|.

Thus unitary operators are the structure-preserving transformations of Hermitian spaces. They are the complex analogue of orthogonal transformations in real inner product spaces.

55.14 Hermitian Orthogonal Projection

Let $W$ be a subspace of a finite-dimensional Hermitian space $V$ . The orthogonal projection of $v$ onto $W$ is the unique vector $p\in W$ such that

v-p\in W^\perp.

q_1,\ldots,q_k

is an orthonormal basis for $W$ , then

p = \sum_{j=1}^k q_j\langle q_j,v\rangle.

In matrix form, with $Q$ having orthonormal columns,

p=QQ^*v.

Thus the projection matrix is

P=QQ^*.

It satisfies

P^2=P, \qquad P^*=P.

So a Hermitian orthogonal projection is an idempotent self-adjoint operator.

55.15 Cauchy-Schwarz Inequality

In a Hermitian space, the Cauchy-Schwarz inequality states that

|\langle u,v\rangle| \le \|u\|\|v\|.

The absolute value is essential because the inner product may be complex.

Equality holds precisely when $u$ and $v$ are linearly dependent.

This inequality implies the triangle inequality for the norm and makes the induced metric well-defined. It also controls correlation, projection, and approximation in complex vector spaces.

55.16 Complex Angles

In a real inner product space, the angle between nonzero vectors is defined using

\cos\theta = \frac{\langle u,v\rangle}{\|u\|\|v\|}.

In a Hermitian space, $\langle u,v\rangle$ may be complex, so this formula does not directly define a real angle.

One common real-valued substitute is

\cos\theta = \frac{|\langle u,v\rangle|}{\|u\|\|v\|}.

This gives a number between $0$ and $1$ . It measures the closeness of the complex lines spanned by $u$ and $v$ .

Orthogonality remains simple:

u\perp v \quad \Longleftrightarrow \quad \langle u,v\rangle=0.

55.17 Hermitian Spaces and Real Inner Product Spaces

Every Hermitian space can be viewed as a real inner product space by taking the real part of the Hermitian product:

(u,v)_{\mathbb{R}} = \operatorname{Re}\langle u,v\rangle.

This is a real inner product on the same underlying set, now regarded as a real vector space.

For example, $\mathbb{C}^n$ has complex dimension $n$ , but real dimension $2n$ . Its Hermitian product contains both the real inner product structure and additional complex structure.

The imaginary part is also meaningful. It is connected to symplectic geometry and quantum mechanics, but the real part gives the ordinary metric geometry.

55.18 Spectral Theorem Preview

Hermitian spaces are the natural setting for the complex spectral theorem.

If $A$ is Hermitian, then there exists a unitary matrix $U$ and a real diagonal matrix $\Lambda$ such that

A=U\Lambda U^*.

The columns of $U$ are orthonormal eigenvectors of $A$ . The diagonal entries of $\Lambda$ are real eigenvalues.

This is the complex version of orthogonal diagonalization for real symmetric matrices. It explains why Hermitian matrices are central in quantum mechanics, signal processing, optimization, and numerical linear algebra.

55.19 Examples

The space $\mathbb{C}^n$ with

\langle x,y\rangle=x^*y

is the standard finite-dimensional Hermitian space.

The space of complex matrices with

\langle A,B\rangle=\operatorname{tr}(A^*B)

is also a Hermitian space.

A space of complex-valued functions on $[a,b]$ may be equipped with

\langle f,g\rangle = \int_a^b \overline{f(t)}g(t)\,dt.

This is the function-space analogue of the standard Hermitian product on $\mathbb{C}^n$ .

Two functions are orthogonal when

\int_a^b \overline{f(t)}g(t)\,dt=0.

This form appears in Fourier analysis and quantum mechanics.

55.20 Summary

A Hermitian space is a complex vector space with a conjugate-symmetric positive definite inner product.

The standard example is

\langle x,y\rangle=x^*y

on $\mathbb{C}^n$ .

The conjugate transpose replaces the transpose throughout complex inner product geometry. Hermitian matrices replace real symmetric matrices. Unitary matrices replace orthogonal matrices. Self-adjoint operators replace symmetric operators.

Hermitian spaces support the same core geometric ideas as real inner product spaces: norm, distance, orthogonality, projection, and orthonormal bases. The additional conjugation is what makes these ideas compatible with complex scalars.

Chapter 55. Hermitian Spaces

55.1 Complex Vector Spaces

55.2 Definition of a Hermitian Inner Product

55.3 The Standard Hermitian Product on Cn\mathbb{C}^nCn

55.4 Norm in a Hermitian Space

55.5 Orthogonality

55.6 Orthonormal Sets

55.7 Coordinates in an Orthonormal Basis

55.8 Hermitian Matrices

55.9 Hermitian Forms

55.10 Positive Definite Hermitian Matrices

55.11 The Adjoint of a Linear Map

55.12 Self-Adjoint Operators

55.13 Unitary Operators

55.14 Hermitian Orthogonal Projection

55.15 Cauchy-Schwarz Inequality

55.16 Complex Angles

55.17 Hermitian Spaces and Real Inner Product Spaces

55.18 Spectral Theorem Preview

55.19 Examples

55.20 Summary

55.3 The Standard Hermitian Product on $\mathbb{C}^n$