In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half". Compare with a bilinear form, which is linear in both arguments; although many authors, especially when working solely in a complex setting, refer to sesquilinear forms as bilinear forms.
Definition and conventions
Conventions differ as to which argument should be linear. We take the first to be conjugate-linear and the second to be linear. This is the convention used by essentially all physicists and originates in Dirac's bra-ket notation in quantum mechanics. The opposite convention is more common in mathematics.
Specifically a map φ : V × V → C is sesquilinear if
for all x,y,z,w ∈ V and all a, b ∈ C.
A sesquilinear form can also be viewed as a complex bilinear map
Given any sesquilinear form φ on V we can define a second sesquilinear form ψ via the conjugate transpose:
In general, ψ and φ will be different. If they are the same then φ is said to be Hermitian. If they are negatives of one another, then φ is said to be skew-Hermitian. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.
Bilinear forms are to squaring (z2), what sesquilinear forms are to Euclidean norm (|z|2 = z*z).
The norm associated to a sesquilinear form is invariant under multiplication by the complex circle (complex numbers of unit norm), while the norm associated to a bilinear form is equivariant (with respect to squaring). Bilinear forms are algebraically more natural, while sesquilinear forms are geometrically more natural.
If B is a bilinear form on a complex vector space and is the associated norm, then .
By contrast, if S is a sesquilinear form on a complex vector space and is the associated norm, then .
- The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain differential form on a Hermitian manifold.
A Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V × V → C such that
The standard Hermitian form on Cn is given (using again the "physics" convention of linearity in the second and conjugate linearity in the first variable) by
A vector space with a Hermitian form (V,h) is called a Hermitian space.
The components of H are given by Hij = h(ei, ej).
The quadratic form associated to a Hermitian form
- Q(z) = h(z,z)
A skew-Hermitian form (also called an antisymmetric sesquilinear form), is a sesquilinear form ε : V × V → C such that
Every skew-Hermitian form can be written as i times a Hermitian form.
The quadratic form associated to a skew-Hermitian form
- Q(z) = ε(z,z)
is always pure imaginary.
A generalization called a semi-bilinear form was used by Reinhold Baer to characterize linear manifolds that are dual to each other in chapter 5 of his book Linear Algebra and Projective Geometry (1952). For a field F and A linear over F he requires
- A pair consisting of an anti-automorphism α of the field F and a function f:A×A→F satisfying
- for all a,b,c ∈ A and
- for all t ∈ F, all x,y ∈ A (page 101)
- (The "transformation exponential notation" is adopted in group theory literature.)
In the algebraic structure called a *-ring the anti-automorphism is denoted by * and forms are constructed as indicated for α. Special constructions such as skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms are all considered in the broader context.
Particularly in L-theory, one also sees the term ε-symmetric form, where , to refer to both symmetric and skew-symmetric forms.