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 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:
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 than 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 × 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 form Q:
This is 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 {ei} for V. The components of B are given by . The quadratic form Q is then given by
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:
- for all a ∈ F and u ∈ V
- Q obeys the parallelogram law:
- The vectors u and v are orthogonal with respect to B iff
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
- for all a ∈ F and u ∈ V, and
- 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.