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sesquilinear forms over general fields

(Definition)

Let be a vector space over a field . may be of any characteristic.

Sesquilinear Forms

Definition 1 A function $b:V\times V\rightarrow k$ is sesquilinear if it satisfies each of the following:

and for all $u,v,w\in V$ ;
For a given field automorphism $\theta$ of , $b(v,lw)=l^\theta b(v,w)$ and for all $v,w\in V$ and $l\in k$ .

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 $\theta=1$ the form is a bilinear form.

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

non-degenerate,
reflexive, (commonly required to define perpendicular);
positive definite (this condition requires the fixed field of $\theta$ , , be an ordered field, such as the rationals $\mathbb{Q}$ or reals $\mathbb{R}$ ).

Non-degenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polarities through the induced $\perp$ operation. (See polarity.)

Hermitian Forms

If $\theta^2=1$ , it is common to exchange notation at this point and use the same notation of $\bar{l}$ for $l^\theta$ as is common for complex conjugation - even if is not $\mathbb{C}$ . Then $\bar{\bar{l}}=l$ .

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

$\displaystyle b(v,w)=\overline{b(w,v)}.$

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.

"sesquilinear forms over general fields" is owned by Algeboy.

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Other names:

Hermitian form, Hermitean form

Also defines:

sesquilinear form, Hermitian form, bilinear form, Hermitean

Keywords:

sesquilinear form, Hermitian form

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Cross-references: Hermitian, even, complex conjugation, point, operation, induced, polarities, dualities, projective geometries, reals, rationals, ordered field, fixed field, positive definite, perpendicular, Reflexive, non-degenerate, properties, combination, variable, automorphism, function, characteristic, field, vector space
There are 12 references to this entry.

This is version 8 of sesquilinear forms over general fields, born on 2006-06-09, modified 2006-06-16.
Object id is 7987, canonical name is SesquilinearFormsOverGeneralFields.
Accessed 3284 times total.

Classification:

AMS MSC:	51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)
	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	47A07 (Operator theory :: General theory of linear operators :: Forms )

Pending Errata and Addenda

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