Quadratic Forms

Abstract

A quadratic form q on E is a map of the form

$$x \to q\left( x \right) = P\left( {x,x} \right)$$

where P is a symmetric bilinear form over E; such a P is well determined by q by the formula

$$P\left( {x,y} \right) = \frac{1}{2}\left( {q\left( {x + y} \right) - q\left( x \right) - \left( y \right)} \right),$$

and is called the polar form of q. Over a subspace F of E, the restriction of q is still a quadratic form, denoted by q| _F .

Here E is a vector space of finite dimension n over a commutative field of characteristic different from 2.