sesquilinear forms over general fields

Let V be a vector spaceMathworldPlanetmath over a field k. k may be of any characteristic.

1 Sesquilinear Forms

Definition 1.

A function b:V×Vk is sesquilinear if it satisfies each of the following:

  1. 1.

    b(v,w+u)=b(v,w)+b(v,u) and b(v+u,w)=b(v,w)+b(u,w) for all u,v,wV;

  2. 2.

    For a given field automorphism θ of k, b(v,lw)=lθb(v,w) and b(lv,w)=lb(v,w) for all v,wV and lk.

Remark 2.

It is possible to apply the field automorphism in the first variable but is more common to do so in the second variable. Also, if θ=1 the form is a bilinear formPlanetmathPlanetmath.

Sesquilinear formsPlanetmathPlanetmath are commonly ascribed any combination of the following properties:

Non-degenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polaritiesMathworldPlanetmathPlanetmath through the induced operation. (See polarity (

2 Hermitian Forms

If θ2=1, it is common to exchange notation at this point and use the same notation of l¯ for lθ as is common for complex conjugation – even if k is not . Then l¯¯=l.

In this notation, Hermitian forms may be defined by the property

Remark 3.

It is not uncommon to see hermitian or Hermitean instead of Hermitian. The name is a tribute to Charles Hermite of the Ecole Polytechnique.

Title sesquilinear forms over general fields
Canonical name SesquilinearFormsOverGeneralFields
Date of creation 2013-03-22 15:58:17
Last modified on 2013-03-22 15:58:17
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 11
Author Algeboy (12884)
Entry type Definition
Classification msc 47A07
Classification msc 15A63
Classification msc 11E39
Classification msc 51A05
Synonym Hermitian form
Synonym Hermitean form
Related topic ReflexiveNonDegenerateSesquilinear
Related topic NonDegenerate
Related topic Polarity2
Related topic ProjectivityMathworldPlanetmath
Related topic ProjectiveGeometry
Related topic Isometry2
Related topic ProjectiveGeometry3
Related topic ClassicalGroups
Defines sesquilinear form
Defines Hermitian form
Defines bilinear form
Defines Hermitean