# Sesquilinear form

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.

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

## Contents

## 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=
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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

where 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

For a fixed *z* in *V* the map is a linear functional on *V* (i.e. an element of the dual space *V**). Likewise, the map is a conjugate-linear functional on *V*.

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.

## Geometric motivation

Bilinear forms are to squaring (*z*^{2}), 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 .

## 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* × *V* → **C** such that

The standard Hermitian form on **C**^{n} is given (using again the "physics" convention of linearity in the second and conjugate linearity in the first variable) by

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 {*e*_{i}} of *V*, a Hermitian form is represented by a Hermitian matrix **H**:

The components of **H** are given by *H*_{ij} = *h*(*e*_{i}, *e*_{j}).

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 *z* ∈ *V*.

## Skew-Hermitian form

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.

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

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*×*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*)*t*(page 101)^{α} - (The "transformation exponential notation"
*t*↦*t*is adopted in group theory literature.)^{α}

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.

## References

- K.W. Gruenberg & A.J. Weir (1977)
*Linear Geometry*, §5.8 Sesquilinear Forms, pp 120–4, Springer, ISBN 0-387-90227-9 . - {{#invoke:citation/CS1|citation

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