Quadratic form - Citizendia

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same In Mathematics, there are several meanings of degree depending on the subject

Quadratic forms are central objects in mathematics, occurring for instance in number theory, Riemannian geometry (as curvature), and Lie theory (via the Killing form). Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Elliptic geometry is also sometimes called Riemannian geometry. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Lie theory is an area of Mathematics, developed initially by Sophus Lie. In Mathematics, the Killing form, named after Wilhelm Killing, is a Symmetric bilinear form that plays a basic role in the theories of Lie groups

They are also ubiquitous in physics and chemistry, as the energy of a system, particularly in relation to the L² norm, which leads to the use of Hilbert spaces. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding This article assumes some familiarity with Analytic geometry and the concept of a limit.

1 Definition
2 Symmetric forms
- 2.1 Details
3 Abstract definition
- 3.1 Away from 2
- 3.2 Further definitions
4 Properties
5 Integral quadratic form
- 5.1 Historical use
- 5.2 Universal quadratic forms
6 Real quadratic forms
7 See also
8 References

Definition

Quadratic forms in one, two, and three variables are given by:

F (x) = a x 2

F (x, y) = a x 2 + b y 2 + c x y

F (x, y, z) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z

Away from 2, quadratic forms are equivalent to symmetric bilinear forms (by the polarization identities), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers. In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric In Mathematics, and more specifically in the theory of Normed spaces and Pre-Hilbert spaces in Functional analysis, a Vector space over the real

The term quadratic form is also often used to refer to a quadratic space, which is a pair (V,q) where V is a vector space over a field k, and q:V → k is a quadratic form on V. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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. Distance is a numerical description of how far apart objects are Three-dimensional space is a geometric model of the physical Universe in which we live

A quadratic form in 2 variables is called a binary quadratic form, and these are extensively studied in number theory (particularly in the theory of modular forms), together with their associated quadratic fields. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and In Mathematics, a quadratic field is an Algebraic number field K of degree two over Q.

Note that general quadratic functions and quadratic equations are not examples of quadratic forms, as they are not always homogeneous: quadratic functions are functions on affine space, while quadratic forms are "functions" on projective space (properly, sections of $\mathcal{O}(2)$ , the square of the twisting sheaf). A quadratic function, in Mathematics, is a Polynomial function of the form f(x=ax^2+bx+c \\! where a \ne 0 \\! In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same In Mathematics, tautological bundle is a term for a particularly natural Vector bundle occurring over a Grassmannian, and more specially over

Any non-zero quadratic form in n variables defines an (n-2)-dimensional quadric in projective space. In Projective geometry a quadric is the set of points of a projective space where a certain Quadratic form on the Homogeneous coordinates becomes zero In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which In this way one may visualize 3-dimensional quadratic forms as conic sections. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

Symmetric forms

When working over a ring where 2 is invertible (for instance, over a field of characteristic not equal to 2), a quadratic form is equivalent to a symmetric bilinear form, in this context often called simply a symmetric form. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric They are thus frequently confused, as in integral quadratic forms (below), or in higher Witt groups. However, they are distinct concepts, and the distinction is frequently important.

Intuitively, a symmetric form generalizes $x y$ , while a quadratic form generalizes $x 2$ , and one can pass between these via the polarization identities. In Mathematics, and more specifically in the theory of Normed spaces and Pre-Hilbert spaces in Functional analysis, a Vector space over the real

Given a quadratic form $Q$ , one obtains a symmetric form $B$ , called the associated symmetric form or associated bilinear form, via:

B (u, v) = Q (u + v) - Q (u) - Q (v)

This corresponds to:

2 x y = (x + y) 2 - x 2 - y 2

Conversely, given a bilinear form $B$ (which need not be symmetric), one obtains a quadratic form via:

Q (u) = B (u, u)

This corresponds to:

$x^2 = x\cdot x$

If one composes these two operations, one gets multiplication by 2 (if one starts with either a quadratic form or a symmetric bilinear form); thus if 2 is invertible, these operations are invertible (the polarization identities); by analogy with

$xy = \frac{1}{2}\left((x+y)^2 - x^2 - y^2\right)$

one takes

$B(u,v) = \frac{1}{2}\left(Q(u+v) - Q(u) - Q(v)\right)$

which gives a 1-1 correspondence between quadratic forms on V and symmetric forms on V. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where In Mathematics, and more specifically in the theory of Normed spaces and Pre-Hilbert spaces in Functional analysis, a Vector space over the real

But if 2 is not invertible, symmetric forms and quadratic forms are different: some quadratic forms cannot be written in the form $B (u, u)$ , for example, over the integers, $Q (u) = x 2 + x y + y 2$ , or more simply $Q (u) = x y$ .

Details

Let us describe this equivalence in the 2 dimensional case. Any 2 dimensional quadratic form may be written as

F (x, y) = a x 2 + b x y + c y 2

Let us write v = (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:

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

Then matrix multiplication gives us the following equality:

F(v)=v^T·M·v

Where the superscript v^T denotes the transpose of a matrix. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a 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 forms. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T}

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. In Mathematics, the real numbers may be described informally in several different ways A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric

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. In Mathematics, the dimension of a Vector space V is the cardinality (i In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The components of B are given by $B i j = B (e i, e j)$ . If 2 is invertible the quadratic form Q is then given by

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

where uⁱ are the components of u in this basis.

Abstract definition

For more details on this topic, see ε-quadratic form.

Let V be a module over a commutative ring R; often R is a field, such as the real numbers, in which case V is a vector space. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

A quadratic form is an element of the symmetric square of the dual space,

$\mbox{Sym}^2\left(V^*\right) := V^* \otimes V^* / \langle v\otimes w - w\otimes v\rangle.$

This is precisely the coordinate-free formulation of "homogeneous degree 2 polynomial", as the symmetric algebra of $V *$ corresponds to polynomials on $V$ . In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals

Bilinear forms are the full tensor product $V^* \otimes V^*$ , and symmetric forms are the subspace of symmetric tensors. In Mathematics, a symmetric tensor is a Tensor that is invariant under a Permutation of its vector arguments Note that the space of quadratic forms is a quotient of the space of bilinear forms, while symmetric forms are a subspace.

In terms of matrices, (we take $V$ to be 2-dimensional):

matrices $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ correspond to bilinear forms
the subspace of symmetric matrices $\begin{pmatrix}a & b\\b & c\end{pmatrix}$ correspond to symmetric forms
the bilinear form $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ yields the quadratic form $a x 2 + b x y + c y x + d y 2 = a x 2 + (b + c) x y + d y 2$ , which is a quotient map with kernel $\begin{pmatrix}0 & b\\-b & 0\end{pmatrix}$ .

One can likewise define quadratic forms corresponding to skew-symmetric forms, Hermitian forms, and skew-Hermitian forms; the general concept is ε-quadratic form. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear

Away from 2

When 2 is invertible in the ring R, one can define a quadratic form in terms of its associated symmetric form in the following way.

A map $Q\colon V \to R$ is called a quadratic form on V if

Q(av) = a² Q(v) for all $a \in R$ and $v \in V$ , and
B(u,v) = Q(u+v) − Q(u) − Q(v) is a bilinear form on V. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where

Here B is called the associated symmetric form; it is a symmetric bilinear form. A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric

Further definitions

Two elements u and v of V are called orthogonal if B(u, v)=0. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

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. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n

A quadratic form Q is called isotropic when there is a non-zero v in V such that $Q (v) = 0$ . In mathematics a Quadratic form over a field F is said to be isotropic if there is a non-zero vector on which it evaluates to zero Otherwise it is called anisotropic. In mathematics a Quadratic form over a field F is said to be isotropic if there is a non-zero vector on which it evaluates to zero A vector or a subspace of a quadratic space may also be referred to as isotropic. If $Q (V) = 0$ then $Q$ is called totally singular.

Properties

Some other properties of quadratic forms:

Q obeys the parallelogram law:

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

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

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

Integral quadratic form

Quadratic forms over the ring of integers are called integral quadratic forms or integral lattices. In Mathematics, the simplest form of the parallelogram law belongs to elementary Geometry. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of They are important in number theory and topology. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

An integral quadratic form is one with integer coefficients, such as $x 2 + x y + y 2$ ; equivalently, given a lattice $Λ$ in a vector space $V$ (over a field with characteristic 0, such as $\mathbf{Q}$ or $\mathbf{R}$ ), a quadratic form $Q$ is integral with respect to $Λ$ if and only if it is integer-valued on $Λ$ , meaning $Q(x,y) \in \mathbf{Z}$ if $x,y \in \Lambda$ .

This is the current use of the term; in the past it was sometimes used differently, as detailed below.

Historical use

Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean:

twos in: the quadratic form associated to a symmetric matrix with integer coefficients
twos out: a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)

This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).

In "twos in", binary quadratic forms are of the form $a x 2 + 2 b x y + c y 2$ , represented by the symmetric matrix $\begin{pmatrix}a & b\\ b&c\end{pmatrix}$ ; this is the convention Gauss uses in Disquisitiones Arithmeticae. The Disquisitiones Arithmeticae is a textbook of Number theory written by German Mathematician Carl Friedrich Gauss in 1798

In "twos out", binary quadratic forms are of the form $a x 2 + b x y + c y 2$ , represented by the symmetric matrix $\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}$ .

Several points of view mean that twos out has been adopted as the standard convention. Those include:

better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
the actual needs for integral quadratic form theory in topology for intersection theory;
the Lie group and algebraic group aspects. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse

Universal quadratic forms

A quadratic form representing all positive integers is sometimes called universal.

Lagrange's four-square theorem shows that $w 2 + x 2 + y 2 + z 2$ is universal. Lagrange's four-square theorem, also known as Bachet's conjecture, was proven in 1770 by Joseph Louis Lagrange.

Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.

Real quadratic forms

Assume $Q$ is a quadratic form defined on a real vector space. In Mathematics, the real numbers may be described informally in several different ways

It is said to be positive definite (resp. In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed negative definite) if $Q (v) > 0$ (resp. $Q (v) < 0$ ) for every vector $v\ne 0.$
If we loosen the strict inequality to ≥ or ≤, the form $Q$ is said to be semidefinite. In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed
If $Q (v) < 0$ for some $v$ and $Q (v) > 0$ for some other $v$ , $Q$ is said to be indefinite. In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed

Let $A$ be the real symmetric matrix associated with $Q$ as described above, so for any column vector $v$ it holds that

Q (v) = v T A v .

Then, $Q$ is positive (semi)definite, negative (semi)definite, indefinite, if and only if the matrix $A$ has the same properties (see positive-definite matrix). In Linear algebra, a positive-definite matrix is a (Hermitian matrix which in many ways is analogous to a Positive Real number. Ultimately, these properties can be characterized in terms of the eigenvalues of $A . In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes$

References

O'Meara, T. If \epsilon is a vector of n Random variables and \Lambda is an n-dimensional symmetric Square matrix, then the In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the (2000), Introduction to Quadratic Forms, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66564-9
Conway, John Horton & Fung, Francis Y. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups C. (1997), The Sensual (Quadratic) Form, Carus Mathematical Monographs, The Mathematical Association of America, ISBN 978-0-88385-030-5

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