A
quadratic form over a field is a map from a finite-dimensional -vector space to such that for all , and the function is a
bilinear form. More concretely, an -ary
quadratic form over a field is a
homogeneous polynomial of degree 2 in variables with coefficients in : q(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K. This formula may be rewritten using matrices: let be the
column vector with components , ..., and be the matrix over whose entries are the coefficients of . Then q(x) = x^\mathsf{T} A x. A vector is a
null vector if . Two -ary quadratic forms and over are
equivalent if there exists a nonsingular linear transformation such that \psi(x) = \varphi(Cx). Let the
characteristic of be different from . The coefficient matrix of may be replaced by the
symmetric matrix with the same quadratic form, so it may be assumed from the outset that is symmetric. Moreover, a symmetric matrix is uniquely determined by the corresponding quadratic form. Under an equivalence , the symmetric matrix of and the symmetric matrix of are related as follows: B = C^\mathsf{T} A C. The
associated bilinear form of a quadratic form is defined by b_q(x,y)=\tfrac{1}{2}(q(x+y)-q(x)-q(y)) = x^\mathsf{T}Ay = y^\mathsf{T}Ax. Thus, is a
symmetric bilinear form over with matrix . Conversely, any symmetric bilinear form defines a quadratic form q(x)=b(x,x), 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 variables are essentially the same.
Quadratic space Given an -dimensional
vector space over a field , a
quadratic form on is a
function that has the following property: for some basis, the function that maps the coordinates of to is a quadratic form. In particular, if with its
standard basis, one has q(v_1,\ldots, v_n)= Q([v_1,\ldots,v_n])\quad \text{for} \quad [v_1,\ldots,v_n] \in K^n. The
change of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in , although the quadratic form depends on the choice of the basis. A finite-dimensional vector space with a quadratic form is called a
quadratic space. The map is a
homogeneous function of degree 2, which means that it has the property that, for all in and in : Q(av) = a^2 Q(v). When the characteristic of is not 2, the bilinear map over is defined: B(v,w)= \tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)). This bilinear form is symmetric. That is, for all , in , and it determines : for all in . When the characteristic of is 2, so that 2 is not a
unit, it is still possible to use a quadratic form to define a symmetric bilinear form . However, can no longer be recovered from this in the same way, since for all (and is thus alternating). Alternatively, there always exists a bilinear form (not in general either unique or symmetric) such that . The pair consisting of a finite-dimensional vector space over and a quadratic map from to is called a
quadratic space, and as defined here is the associated symmetric bilinear form of . The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, is also called a quadratic form. Two -dimensional quadratic spaces and are
isometric if there exists an invertible linear transformation (
isometry) such that Q(v) = Q'(Tv) \text{ for all } v\in V. The isometry classes of -dimensional quadratic spaces over correspond to the equivalence classes of -ary quadratic forms over .
Generalization Let be a
commutative ring, be an -
module, and be an -bilinear form. A mapping is the
associated quadratic form of , and is the
polar form of . A quadratic form may be characterized in the following equivalent ways: • There exists an -bilinear form such that is the associated quadratic form. • for all and , and the polar form of is -bilinear.
Related concepts Two elements and of are called
orthogonal if . The
kernel of a bilinear form consists of the elements that are orthogonal to every element of . is
non-singular if the kernel of its associated bilinear form is . If there exists a non-zero in such that , the quadratic form is
isotropic, otherwise it is
definite. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of to a subspace of is identically zero, then is
totally singular. The orthogonal group of a non-singular quadratic form is the group of the linear automorphisms of that preserve : that is, the group of isometries of into itself. If a quadratic space has a product so that is an
algebra over a field, and satisfies \forall x, y \isin A \quad Q(x y) = Q(x) Q(y), then it is a
composition algebra. == Equivalence of forms ==