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:

$F(x)=ax^{2}$

$F(x,y)=ax^{2}+by^{2}+cxy$

$F(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz$

Note that general quadratic functions and quadratic equations are not examples of quadratic forms.

Symmetric bilinear forms

In most cases (i.e. where the characteristic of the underlying field is not 2) quadratic forms may be equivalently viewed as symmetric bilinear forms. Let us describe this equivalence in the 2 dimensional case. Any 2 dimensional quadratic form may be written as

$F(x,y)=ax^{2}+by^{2}+cxy$.

Let us write x = (x,y) for any vector in the vector space. The quadratic form F can be expressed in terms of matrices if we let M be the 2×2 matrix:

$M={\begin{bmatrix}a&c/2\\c/2&b\end{bmatrix}}.$

Then matrix multiplication gives us the following equality:

F(x)=x^T·M·x

Where the superscript x^T denotes the transpose of a matrix. Notice we have used that the characteristic is not 2, since we divided by 2 to define M. So we see the correspondence between 2 dimensional quadratic forms F and 2×2 symmetric matrices M, which correspond to symmetric bilinear forms.

This observation generalises quickly to forms in n variables and n×n symmetric matrices. For example, in the case of real-valued quadratic forms, the characteristic of the real numbers is 0, so real quadratic forms and real symmetric bilinear forms are the same objects, from different points of view.

It has long been known, particularly from some aspects of number theory, that this is not the complete story. In fact there has been, historically speaking, some controversy over whether the notion of integral quadratic form should be presented with twos in (i.e., based on integral symmetric matrices) or twos out. In the notation above, therefore, the controversy is whether the term integral should imply a,b, and c are integers, or whether it should imply a, b, and c/2 are integers.

Several points of view mean that twos out has been adopted as the standard convention. Those include: (i) better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty; (ii) the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s; (iii) the actual needs for integral quadratic form theory in topology for intersection theory; and (iv) the Lie group and algebraic group aspects.

The rest of this article proceeds with the accepted way to handle the issue, which therefore has particular relevance to working over some ring R in which 2 is not a unit.

Quadratic form on a module or vector space

Let V be a module over a commutative ring F; often V is a vector space over a field F.

A map Q : V → F is called a quadratic form on V if

• Q(au) = a² Q(u) for all a ∈ F and u ∈ V, and
• B(u,v) = Q(u+v) − Q(u) − Q(v) is a bilinear form on V.

B is called the associated bilinear form. Note that for any vector u ∈ V

2Q(u) = B(u,u)

so if 2 is invertible in F we can recover the quadratic form from the symmetric bilinear form B by

Q(u) = B(u,u)/2.

When 2 is invertible this gives a 1-1 correspondence between quadratic forms on V and symmetric bilinear forms on V. If B is any symmetric bilinear form then B(u,u) is always a quadratic form. This is sometimes used as the definition of a quadratic form, but if 2 is not invertible this way of writing the definition is wrong: some quadratic forms cannot be written in the form B(u,u).

Quadratic forms over the ring of integers are called integral quadratic forms or integral lattices. They are important in number theory and topology.

Two elements u and v of V are called orthogonal if B(u, v)=0.

The kernel of the bilinear form B consists of the elements that are orthogonal to all elements of V, and the kernel of the quadratic form Q consists of all elements u of the kernel of B with Q(u)=0. If 2 is invertible then Q and its associated bilinear form B have the same kernel.

The bilinear form B is called non-singular if its kernel is 0, and the quadratic form Q is called non-singular if its kernel is 0.

The orthogonal group of a non-singular quadratic form Q is the group of automorphisms of V that preserve the quadratic form Q.

If V is free of rank n we write the bilinear form B as a symmetric matrix B relative to some basis {e_i} for V. The components of B are given by $B_{ij}=B(e_{i},e_{j})$. If 2 is invertible the quadratic form Q is then given by

$2Q(u)=\mathbf {u} ^{T}\mathbf {Bu} =\sum _{i,j=1}^{n}B_{ij}u^{i}u^{j}$

where uⁱ are the components of u in this basis.

Some other properties of quadratic forms:

• Q obeys the parallelogram law:

$Q(u+v)+Q(u-v)=2Q(u)+2Q(v)$

• The vectors u and v are orthogonal with respect to B if and only if

$Q(u+v)=Q(u)+Q(v)$

Definiteness of a quadratic form

Assume Q is a quadratic form defined on a real vector space.

• It is said to be positive definite (resp. negative definite) if Q(v) > 0 (resp. Q(v) < 0) for every vector v ≠ 0.
• If we loosen the strict inequality to ≥ or ≤, the form Q is said to be semidefinite.

Assume that a quadratic form is represented as a matrix.

• The matrix is positive (resp. negative) definite if and only if it has all strictly positive (resp. negative) eigenvalues.
• Similarly, the matrix is positive (resp. negative) semidefinite if and only if it has all non-negative (resp. non-positive) eigenvalues.

See symmetric matrix for a justification of this statement.

Now assume that a quadratic form is represented by a real symmetric matrix. In fact, this is possible for any quadratic form with real coefficients.

• The real symmetric matrix is positive (resp. negative) definite if and only if it has all strictly positive (resp. negative) leading principal minors.
• Similarly, the real symmetric matrix is positive (resp. negative) semidefinite if and only if it has all non-negative (resp. non-positive) leading principal minors.

As minors are easy to compute and any quadratic with real coefficients has a symmetric matrix representation, this is a common quick way to check for definiteness.

See equivalent formulation of positive-definite matrices for more information about positive definiteness of Hermitian matrices, which are generalizations of symmetric matrices.

Isotropic Spaces

A quadratic form Q is called isotropic when there is a non-zero v in V such that $Q(v)=0$. Otherwise it is called anisotropic. If $Q(V)=0$ then $Q$ is called totally isotropic.

See also

• Quadratic form (statistics)

References

• O'Meara, T. (2000). Introduction to Quadratic Forms. Berlin, Heidelberg: Springer-Verlag. ISBN 3-540-66564-1.