Hermitian form over a division ring

Let D be a division ring admitting an involutionMathworldPlanetmath (http://planetmath.org/Involution2) *. Let V be a vector spaceMathworldPlanetmath over D. A Hermitian formMathworldPlanetmathPlanetmath over D is a function from V×V to D, denoted by (,) with the following properties, for any v,wV and dD:

  1. 1.

    (,) is additive in each of its arguments,

  2. 2.


  3. 3.


  4. 4.


Note that if the Hermitian form (,) is non-trivial and if * is the identity on D, then D is a field and (,) is just a symmetric bilinear formMathworldPlanetmath.

If we replace the last condition by (u,v)=-(v,u)*, then (,) over D is called a skew Hermitian form.

Remark. Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.

Title Hermitian form over a division ring
Canonical name HermitianFormOverADivisionRing
Date of creation 2013-03-22 15:41:04
Last modified on 2013-03-22 15:41:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 15A63
Defines Hermitian form
Defines skew Hermitian form