# Sesquilinear form

In mathematics, a sesquilinear form on a complex vector space V is a map V × VC 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.

A motivating example is the inner product on a complex vector space, which is not bilinear, but instead sesquilinear. See geometric motivation below.

## 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{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} and originates in Dirac's bra–ket notation in quantum mechanics. The opposite convention is more common in mathematics{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}.

Specifically a map φ : V × VC is sesquilinear if

{\begin{aligned}&\phi (x+y,z+w)=\phi (x,z)+\phi (x,w)+\phi (y,z)+\phi (y,w)\\&\phi (ax,by)={\bar {a}}b\,\phi (x,y)\end{aligned}} for all x,y,z,wV and all a, bC. ${\bar {a}}$ is the complex conjugate of a.

A sesquilinear form can also be viewed as a complex bilinear map

${\bar {V}}\times V\to {\mathbf {C} }$ where ${\bar {V}}$ is the complex conjugate vector space to V. By the universal property of tensor products these are in one-to-one correspondence with (complex) linear maps

${\bar {V}}\otimes V\to {\mathbf {C} }.$ Given any sesquilinear form φ on V we can define a second sesquilinear form ψ via the conjugate transpose:

$\psi (w,z)={\overline {\varphi (z,w)}}.$ 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.

## Geometric motivation

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.

By contrast, if S is a sesquilinear form on a complex vector space and $|x|_{S}:=S(x,x)$ is the associated norm, then $|ix|_{S}=S(ix,ix)={\bar {i}}iS(x,x)=|x|_{S}$ .

## Hermitian form

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 × VC such that

$h(w,z)={\overline {h(z,w)}}.$ 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

$\langle w,z\rangle =\sum _{i=1}^{n}{\overline {w_{i}}}z_{i}.$ More generally, the inner product on any complex Hilbert space is a Hermitian form.

A vector space with a Hermitian form (V,h) is called a Hermitian space.

If V is a finite-dimensional space, then relative to any basis {ei} of V, a Hermitian form is represented by a Hermitian matrix H:

$h(w,z)={\overline {\mathbf {w} ^{\mathrm {T} }}}\mathbf {Hz} .$ The components of H are given by Hij = h(ei, ej).

The quadratic form associated to a Hermitian form

Q(z) = h(z,z)

is always real. Actually, one can show that a sesquilinear form is Hermitian iff the associated quadratic form is real for all zV.

## Skew-Hermitian form

A skew-Hermitian form (also called an antisymmetric sesquilinear form), is a sesquilinear form ε : V × VC such that

$\varepsilon (w,z)=-{\overline {\varepsilon (z,w)}}.$ Every skew-Hermitian form can be written as i times a Hermitian form.

If V is a finite-dimensional space, then relative to any basis {ei} of V, a skew-Hermitian form is represented by a skew-Hermitian matrix A:

$\varepsilon (w,z)={\overline {\mathbf {w} }}^{\mathrm {T} }\mathbf {Az} .$ The quadratic form associated to a skew-Hermitian form

Q(z) = ε(z,z)

is always pure imaginary.

## Generalization

A generalization called a Template:Visible anchor 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 × AF satisfying
for all a,b,cA: f(a + b, c) = f(a, c) + f(b, c), f(a, b + c) = f(a, b) + f(a, c), and
for all tF and x,yA: f(tx, y) = tf(x, y), f(x, ty) = f(x, y) tα (page 101)
(The "transformation exponential notation" ttα is adopted in group theory literature.)

Baer calls such a form an α-form over A. The usual sesquilinear form has complex conjugation for α. When α is the identity, then f is a bilinear form.

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.