# 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.

Two variables:

Three variables:

## 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.

See also: