Quadratic Form - Definitions

Definitions

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 x₁, …, x_n and A = (a_ij) be the n×n matrix over K whose entries are the coefficients of q. Then

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

Let us assume that the characteristic of K is different from 2. (The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems have to be modified.) The coefficient matrix A of q may be replaced by the symmetric matrix (A + AT)/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 T, 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_q 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 one another. 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.