sesquilinear forms over general fields
Let be a vector space over a field . may be of any characteristic.
1 Sesquilinear Forms
A function is sesquilinear if it satisfies each of the following:
and for all ;
For a given field automorphism of , and for all and .
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 the form is a bilinear form.
Sesquilinear forms are commonly ascribed any combination of the following properties:
reflexive, (commonly required to define perpendicular);
positive definite (this condition requires the fixed field of , , be an ordered field, such as the rationals or reals ).
Non-degenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polarities through the induced operation. (See polarity (http://planetmath.org/Polarity2).)
2 Hermitian Forms
If , it is common to exchange notation at this point and use the same notation of for as is common for complex conjugation – even if is not . Then .
In this notation, Hermitian forms may be defined by the property
|Title||sesquilinear forms over general fields|
|Date of creation||2013-03-22 15:58:17|
|Last modified on||2013-03-22 15:58:17|
|Last modified by||Algeboy (12884)|