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Hermitian form over a division ring
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(Definition)
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Let $D$ be a division ring admitting an involution $*$ Let $V$ be a vector space over $D$ A Hermitian form over $D$ is a function from $V\times V$ to $D$ denoted by $(\cdot,\cdot)$ with the following properties, for any $v,w\in V$ and $d\in D$
- $(\cdot,\cdot)$ is additive in each of its arguments,
- $(du,v)=d(u,v)$
- $(u,dv)=(u,v)d^*$
- $(u,v)=(v,u)^*$
Note that if the Hermitian form $(\cdot,\cdot)$ is non-trivial and if $*$ is the identity on $D$ then $D$ is a field and $(\cdot,\cdot)$ is just a symmetric bilinear form.
If we replace the last condition by $(u,v)=-(v,u)^*$ then $(\cdot,\cdot)$ over $D$ is called a skew Hermitian form.
Remark. Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.
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"Hermitian form over a division ring" is owned by CWoo.
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| Also defines: |
Hermitian form, skew Hermitian form |
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Cross-references: induces, symmetric bilinear form, field, identity, arguments, additive, properties, function, vector space, division ring
This is version 9 of Hermitian form over a division ring, born on 2006-02-16, modified 2006-10-22.
Object id is 7626, canonical name is HermitianFormOverADivisionRing.
Accessed 2847 times total.
Classification:
| AMS MSC: | 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products) |
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Pending Errata and Addenda
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