Quadratic form - Definition and Overview

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 three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six variables, the three coordinates of each of the two points.

Quadratic forms in one, two, and three variables are given by:

<math>F(x) = ax^2<math>

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

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

Quadratic form on a vector space

Let V be a vector space V over a field F. For now we assume that F has characteristic different from 2. This is true, in particular, for the real and complex number fields which have characteristic 0. The case char(F) = 2 is somewhat exceptional, and will be treated separately.

A map Q : V → F is called a quadratic form on a V if there exists a symmetric bilinear form B : V &times V → F such that

Q(u) = B(u,u) for all u ∈ V

B is called the associated bilinear form. Note that for any vectors u,v ∈ V

Q(u+v) = Q(u) + 2B(u,v) + Q(v)

so we can recover the bilinear form B from Q:

<math>B(u,v) = \frac{1}{2}\left(Q(u+v) - Q(u) - Q(v)\right)<math>

This is an example of polarization of an algebraic form. There is then a 1-1 correspondence between quadratic forms on V and symmetric bilinear forms on V. Given one we can uniquely define the other.

If V has dimension n we write the bilinear form B as a symmetric matrix B relative to some basis {e_i} for V. The components of B are given by <math>B_{ij} = B(e_i,e_j)<math>. The quadratic form Q is then given by

<math>Q(u) = \mathbf{u}^T \mathbf{Bu} = \sum_{i,j=1}^{n}B_{ij}u^i u^j<math>

where uⁱ are the components of u in this basis. Note that Q(u) is a homogeneous polynomial of degree two in the coordinates of u and so agrees with our original definition.

Some other properties of quadratic forms:

• <math>Q(au) = a^2 Q(u)<math> for all a ∈ F and u ∈ V
• Q obeys the parallelogram law:

<math>Q(u+v) + Q(u-v) = 2Q(u) + 2Q(v)<math>

• The vectors u and v are orthogonal with respect to B iff

<math>Q(u+v) = Q(u) + Q(v)<math>

Characteristic two

The theory of quadratic forms in characteristic two has quite a different flavor, essentially because division by 2 is not possible. It is no longer true that every quadratic form is of the form Q(u) = B(u,u) for a symmetric bilinear form B. Moreover, even if B exists it is not unique: since alternating forms are also symmetric in characteristic two, one can add any alternating form to B and get the same quadratic form.

A more general definition of a quadratic form which works for any characteristic is as follows. A quadractic form on a vector space V over a field F is as a map Q : V → F such that

• <math>Q(au) = a^2 Q(u)<math> for all a ∈ F and u ∈ V, and
• <math>Q(u+v) - Q(u) - Q(v)<math> is a bilinear form on V.

Generalizations

One can generalize the notion of a quadratic form to modules over a commutative ring. Integral quadratic forms are important in number theory and topology.