Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, the distance between two points in space is found by taking the square root of a quadratic form involving six variables, the three co-ordinates of each of the two points.

Examples

Two variables:

<math>F(x,y) = ax^2 + by^2 + 2cxy</math>

Three variables:

<math>F(x,y,z) = ax^2 + by^2 + cz^2 + 2dxy + 2exz + 2fyz</math>

Relation with bilinear forms

To express the quadratic form concept in linear algebra terms, we can note that for any bilinear form B on a vector space V of finite dimension, the expression B(v,v) for v in V will be a quadratic form in the co-ordinates of v with respect to a fixed basis. If F is the underlying field, then this is in fact the general quadratic form over F, unless the characteristic of F is 2. Provided we can divide by 2 in F there is no problem in writing down a matrix representing B, to give rise to any fixed quadratic form: we can choose B to be symmetric.

In fact under that condition there is a 1-1 correspondence between quadratic forms Q and symmetric bilinear forms B (an example of polarisation). For the purposes of quadratic form theory over rings in general, such as the integral quadratic forms important in number theory and topology, one must start with a more careful definition to avoid problems caused by division by 2.