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 Latin numerical prefix sesqui- meaning "one and a half". Compare with a bilinear form, which is linear in both arguments. However 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 (i.e. antilinear) 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. is the complex conjugate of a.
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: f(a + b, c) = f(a, c) + f(b, c), f(a, b + c) = f(a, b) + f(a, c), and
- for all t ∈ F and x,y ∈ A: f(tx, y) = tf(x, y), f(x, ty) = f(x, y) tα (page 101)
- (The "transformation exponential notation" t ↦ tα 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 ε = ±1, to refer to both symmetric and skew-symmetric forms.