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sesquilinear forms over general fields
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(Definition)
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Let $V$ be a vector space over a field $k$ . $k$ may be of any characteristic.
Definition 1 A function $b:V\times V\rightarrow k$ is sesquilinear if it satisfies each of the following:
- $b(v,w+u)=b(v,w)+b(v,u)$ and $b(v+u,w)=b(v,w)+b(u,w)$ for all $u,v,w\in V$ ;
- For a given field automorphism $\theta$ of $k$ , $b(v,lw)=l^\theta b(v,w)$ and $b(lv,w)=lb(v,w)$ 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 sesquilinear and bilinear forms apply to projective geometries as dualities and polarities through the induced $\perp$ operation. (See polarity.)
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 $k$ is not $\mathbb{C}$ . Then $\bar{\bar{l}}=l$ .
In this notation, Hermitian forms may be defined by the property $$ b(v,w)=\overline{b(w,v)} $$
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.
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"sesquilinear forms over general fields" is owned by Algeboy.
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See Also: reflexive non-degenerate sesquilinear, non-degenerate, polarity, projectivity, projective geometry, isometry, projective geometry, classical groups
| Other names: |
Hermitian form, Hermitean form |
| Also defines: |
sesquilinear form, Hermitian form, bilinear form, Hermitean |
| Keywords: |
sesquilinear form, Hermitian form |
This object's parent.
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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 13 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 5715 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 ) |
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Pending Errata and Addenda
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