structural:expansion
Definitions

A quadratic form over a field K is a map q : V → K from a finite-dimensional K -vector space to K such that q (av) = a q (v) for all a ∈ K, v ∈ V and the function q (u + v) − q (u) − q (v) is bilinear.

More concretely, an n -ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K:

This formula may be rewritten using matrices: let x be the column vector with components x,..., x and A = (a) be the n × n matrix over K whose entries are the coefficients of q. Then

A vector v = (x,..., x) is a null vector if q (v) = 0.

Two n -ary quadratic forms φ and ψ over K are equivalent if there exists a nonsingular linear transformation C ∈ GL (n, K) such that

Let the characteristic of K be different from 2. The coefficient matrix A of q may be replaced by the symmetric matrix (A + A)/2 with the same quadratic form, so it may be assumed from the outset that A is symmetric. Moreover, a symmetric matrix A is uniquely determined by the corresponding quadratic form. Under an equivalence C, the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows:

The associated bilinear form of a quadratic form q is defined by

Thus, b is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form

and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.