1 Definition and basic properties
A Hermitian form is a function on a complex vector space that assigns a complex number to each ordered pair of vectors. It is designed to behave like the usual dot product, but adapted to complex scalars. The form is linear in one argument, conjugate-linear in the other, and symmetric up to complex conjugation. Because of these features, Hermitian forms serve as the basic algebraic framework for complex geometry, inner product spaces, and many constructions in linear algebra.
1.1 Sesquilinearity
A Hermitian form is sesquilinear, meaning it is linear in one variable and conjugate-linear in the other. If the first variable is chosen to be linear, then for vectors x, y, z and complex numbers a, b, one has
h(ax + by, z) = a h(x, z) + b h(y, z)
and
h(x, ay + bz) = \u0304a h(x, y) + \u0304b h(x, z).
This mixed linear behavior distinguishes Hermitian forms from bilinear forms over the real numbers. It is the reason complex conjugation appears throughout the theory.
1.2 Conjugate symmetry
A Hermitian form satisfies conjugate symmetry:
h(x, y) = \u0304h(y, x).
This condition implies that the value of the form on a vector with itself is always real, since
h(x, x) = \u0304h(x, x).
Conjugate symmetry is the complex analogue of symmetry for real inner products and ensures that the form encodes a balanced notion of angle and length.
1.3 Associated quadratic form
Every Hermitian form determines a real-valued quadratic expression by evaluating the form on a vector and itself. For a vector x, the quantity h(x, x) is real, and in many settings it is used to measure length or energy. When the form is positive definite, this quantity is strictly positive for nonzero vectors and becomes the square of a norm.
The associated quadratic form is often the most immediate way to extract geometric information from a Hermitian form. It can reveal whether the form is positive, indefinite, or degenerate.
1.4 Nondegeneracy
A Hermitian form is nondegenerate if the only vector orthogonal to every vector is the zero vector. Equivalently, if h(x, y) = 0 for all y implies x = 0. Nondegeneracy means that the form provides a faithful pairing between the vector space and its dual-type structure.
Nondegenerate forms are especially important because they allow the transfer of information between vectors, linear functionals, and operators. Degenerate forms, by contrast, have a nontrivial null space and are less rigid.
2 Matrix representation
Hermitian forms on finite-dimensional complex vector spaces can be represented by matrices once a basis is chosen. This makes them computationally accessible and connects the abstract definition to familiar matrix algebra. The matrix representation also clarifies how the form changes under coordinate transformations.
2.1 Hermitian matrices
In a chosen basis, a Hermitian form is represented by a Hermitian matrix, that is, a matrix equal to its own conjugate transpose. If A is the matrix of the form, then A = A*, where A* denotes the conjugate transpose.
Such matrices have real diagonal entries, and their off-diagonal entries occur in conjugate pairs. This matrix condition is the coordinate form of conjugate symmetry.
2.2 Change of basis
When the basis changes, the matrix representing the form changes by a congruence transformation rather than by similarity. If P is the matrix of the basis change, the new matrix is typically P*AP. This reflects the fact that the form acts on two vector arguments, one of which is conjugated.
Because of this transformation law, many properties of Hermitian forms remain unchanged under change of basis. These include rank, signature-type data, and degeneracy.
2.3 Coordinate formulas
If x and y are vectors with coordinate column vectors v and w in a chosen basis, and A is the matrix of the Hermitian form, then
h(x, y) = v*Aw.
This compact formula is the standard computational expression for a Hermitian form. It shows how the form can be evaluated from coordinates using matrix multiplication and conjugate transpose.
2.4 Rank and determinant
The rank of the representing matrix measures the size of the nondegenerate part of the form. Full rank corresponds to nondegeneracy in finite dimensions. The determinant is nonzero exactly when the form is nondegenerate.
These invariants are useful for classification. They remain tied to the intrinsic structure of the form, even though the matrix itself depends on the chosen basis.
3 Classification
Hermitian forms are classified according to the behavior of the value h(x, x). The principal cases are positive definite, indefinite, and degenerate forms. This classification determines the geometry induced by the form and the kind of canonical basis one can choose.
3.1 Positive definite Hermitian forms
A Hermitian form is positive definite if h(x, x) > 0 for every nonzero vector x. This is the case that underlies the standard notion of a complex inner product. Positive definite forms generate norms, angles, and orthogonality in a way closely analogous to Euclidean geometry.
Such forms are the most rigid and best behaved. In finite-dimensional spaces, they can be simplified by choosing an orthonormal basis.
3.2 Indefinite Hermitian forms
A Hermitian form is indefinite if it takes both positive and negative values on vectors of nonzero length. In this case, the form does not define a norm in the usual sense, but it still encodes meaningful geometric structure.
Indefinite Hermitian forms appear in settings where one needs to distinguish directions of different type or sign. Their study often involves separating the space into subspaces on which the form is positive and negative.
3.3 Degenerate forms
A Hermitian form is degenerate if there exists a nonzero vector x such that h(x, y) = 0 for all y. Such a vector lies in the radical, or null space, of the form. Degeneracy means that the form fails to detect some directions in the vector space.
Degenerate forms arise naturally in quotient constructions and in situations where a pairing is introduced before removing redundancy. They are less suitable for geometry, but still important in algebraic contexts.
3.4 Canonical forms
Over a finite-dimensional complex vector space, Hermitian forms can often be reduced to canonical diagonal forms by a suitable choice of basis. In the nondegenerate case, a form can be arranged into a diagonal matrix with entries equal to 1 or -1 after rescaling basis vectors.
This reduction provides a standard classification by the numbers of positive and negative directions, together with any zero directions in the degenerate case. Canonical forms make the structure of the space transparent.
4 Relationship to inner product spaces
Hermitian forms are the algebraic foundation of complex inner product spaces. When a Hermitian form is positive definite, it becomes an inner product and supports the usual analytic and geometric tools. The resulting structures are central in functional analysis, quantum theory, and numerical linear algebra.
4.1 Hermitian inner products
A Hermitian inner product is a positive definite Hermitian form. It satisfies the same sesquilinear and conjugate symmetry conditions, along with positivity. This is the complex counterpart of the real dot product.
Inner products allow one to measure lengths and angles in complex vector spaces. They are the starting point for many standard constructions, including projection and orthogonal decomposition.
4.2 Norms and distances
From a Hermitian inner product, one defines a norm by
| x | = sqrt(h(x, x)). |
|---|
This norm then induces a distance between vectors by
| d(x, y) = | x - y | . |
|---|
These notions make complex vector spaces into metric spaces with a rich geometric structure. Many familiar inequalities and convergence concepts follow from this setup.
4.3 Orthogonality
Two vectors are orthogonal if their Hermitian inner product is zero. Orthogonality is one of the most useful features of Hermitian forms because it separates a space into independent components.
Orthogonal decompositions simplify calculations, reveal structure, and support methods such as least squares approximation. In the positive definite case, orthogonality behaves in a particularly stable way.
4.4 Orthonormal bases
An orthonormal basis is a basis whose vectors are mutually orthogonal and each have norm 1. In finite-dimensional inner product spaces, such bases always exist. They are especially convenient because the matrix of the inner product in an orthonormal basis is the identity matrix.
Orthonormal bases simplify coordinate calculations, projections, and operator matrices. They are a standard tool in both pure and applied mathematics.
5 Linear transformations
Hermitian forms interact closely with linear operators on complex vector spaces. They provide a way to define adjoints, identify special classes of operators, and study transformations that preserve geometric structure.
5.1 Adjoint operators
Given a Hermitian inner product, the adjoint of a linear operator T is the operator T* satisfying
h(Tx, y) = h(x, T*y)
for all vectors x and y. The adjoint is defined relative to the chosen Hermitian form and generalizes the transpose of a real matrix.
Adjoints are fundamental in operator theory because they encode how an operator interacts with the geometry of the space. They are also central to the study of normal, self-adjoint, and unitary operators.
5.2 Self-adjoint operators
An operator is self-adjoint if it equals its adjoint. Such operators are the complex analogue of symmetric real matrices. They have especially strong spectral properties and often arise from physically meaningful quantities.
Self-adjoint operators are closely tied to Hermitian forms because their matrix representations are Hermitian with respect to an orthonormal basis. This connection makes them a major object of study in linear algebra.
5.3 Unitary transformations
A unitary transformation preserves the Hermitian inner product. Equivalently, it preserves lengths, angles, and orthogonality. In matrix terms, a unitary matrix U satisfies U*U = I.
Unitary transformations are the natural isometries of complex inner product spaces. They are indispensable in harmonic analysis, quantum mechanics, and numerical computation.
5.4 Invariance under basis change
Many properties associated with Hermitian forms are invariant under a change of basis that respects the underlying vector space structure. The form itself changes coordinates, but geometric features such as nondegeneracy and signature-type data remain the same.
This invariance is one reason Hermitian forms are so useful: they describe intrinsic structure rather than artifacts of a particular coordinate system. Basis change often reveals the simplest representation of that structure.
6 Geometry and applications
Hermitian forms appear in geometric settings wherever complex coordinates and metric ideas meet. They also provide a language for many applied theories, especially those involving oscillation, energy, and symmetry. Their influence extends from classical geometry to modern mathematical physics.
6.1 Complex Euclidean geometry
In complex Euclidean geometry, Hermitian forms play the role that dot products play in real Euclidean space. They support notions of orthogonality, length, and angle in complex coordinates.
Although complex spaces do not behave exactly like real Euclidean spaces, Hermitian forms give them a compatible geometric structure. This is essential for many constructions in complex linear algebra and geometry.
6.2 Quadratic spaces over complex numbers
Hermitian forms can be viewed as a complex analogue of quadratic spaces, with the important difference that the values are constrained by conjugation. They organize vector spaces according to the sign and degeneracy of the associated quadratic expression.
This viewpoint is helpful in classification problems, where one studies the number and type of directions on which the form is positive, negative, or zero. It also clarifies the role of isotropic vectors, which satisfy h(x, x) = 0.
6.3 Use in physics
Hermitian forms are widespread in physics because many observables and energy-like quantities are naturally represented by Hermitian structures. In quantum theory, inner products determine probabilities and transition amplitudes, while Hermitian operators model measurable quantities.
Their role is not limited to quantum mechanics. They also appear in classical wave theory, signal analysis, and any setting where complex amplitudes are combined with conservation or orthogonality principles.
6.4 Use in differential geometry
In differential geometry, Hermitian forms contribute to the study of complex manifolds and complex vector bundles. They provide a way to measure tangent vectors in a manner compatible with complex structure.
When combined with smooth variation from point to point, Hermitian forms yield Hermitian metrics. These metrics are central in complex differential geometry and help connect analytic and geometric methods.
7 Examples
Concrete examples make the abstract definitions easier to interpret. The most familiar examples come from standard coordinate spaces, but indefinite and degenerate cases also illustrate the range of behavior possible for Hermitian forms.
7.1 Standard Hermitian form on C^n
The standard Hermitian form on C^n is
h(x, y) = x1\u0304y1 + x2\u0304y2 + ... + xn\u0304yn,
where x and y are vectors with complex coordinates. This is the usual complex inner product, and it is positive definite.
It induces the standard norm and orthogonality relation on C^n. In the standard basis, its matrix is the identity matrix.
7.2 Indefinite example
An example of an indefinite Hermitian form on C^2 is
h(x, y) = x1\u0304y1 - x2\u0304y2.
Here the form takes positive values on vectors with large first coordinate and negative values on vectors with large second coordinate. It is nondegenerate but not positive definite.
This example shows that Hermitian forms need not define a norm. Instead, they can split the space into directions of different sign.
7.3 Degenerate example
A degenerate Hermitian form on C^2 is given by
h(x, y) = x1\u0304y1.
In this case, vectors with first coordinate 0 are orthogonal to every vector. The form therefore has a nontrivial radical and is not nondegenerate.
This example illustrates how a Hermitian form can ignore part of the space. Such forms are useful in contexts where one later factors out the null directions.