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 form on a module or vector space

Let V be a module V over a commutative ring F;

Related Topics:
Module - Commutative ring

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often V is a vector space over a field F.

Related Topics:
Vector space - Field

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

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• 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

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

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so if 2 is invertible in F we can recover the quadratic form from the

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symmetric bilinear form B by

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

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When 2 is invertible

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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 definition is wrong as not all quadratic forms can be obtained like this.

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Quadratic forms over the ring of integers are called integral quadratic forms or integral lattices.

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They are important in number theory and topology.

Related Topics:
Number theory - Topology

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

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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.

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If 2 is invertible then Q and its associated bilinear form B have the same kernel.

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The bilinear form B is called non-singular if its kernel is 0,

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and the quadratic form Q is called non-singular if its kernel is 0.

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

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If V is free of rank n we write the bilinear form B as a symmetric matrix B relative to some basis {ei} 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

Related Topics:
Rank - Symmetric matrix - Basis

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:2 Q(u) = mathbf{u}^T mathbf{Bu} = sum_{i,j=1}^{n}B_{ij}u^i u^j

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where ui are the components of u in this basis.

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Some other properties of quadratic forms:

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• Q obeys the parallelogram law:

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

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

::Q(u+v) = Q(u) + Q(v)

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