Quadratic form

A quadratic form over a field $K$ is a map $q : V \to K$ from a finite-dimensional vector space $V$ over $K$ such that

q (a v) = a^{2} q (v), a \in K, v \in V

and the function

b_{q} (u, v) := \frac{1}{2} (q (u + v) - q (u) - q (v))

is bilinear. It takes the form of a degree 2 homogeneous polynomial in $n$ variables with coefficients in $K$ .

Fixed a basis of $V$ , there exists a $n \times n$ matrix $A$ such that

q (x) = x^{T} A x

The bilinear map $b_{q}$ defined above is called the associated bilinear form. It is satisfied that

q (u, v) = v^{T} A u

and

q (x) = b_{q} (x, x) .