Isotropic quadratic form

Last updated

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form.


Suppose that (V, q) is quadratic space and W is a subspace. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors in it are isotropic, and an anisotropic subspace if it does not contain any (non-zero) isotropic vectors. The isotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. [1]

A quadratic form q on a finite-dimensional real vector space V is anisotropic if and only if q is a definite form:

  • either q is positive definite, i.e. q(v) > 0 for all non-zero v in V ;
  • or q is negative definite, i.e. q(v) < 0 for all non-zero v in V.

More generally, if the quadratic form is non-degenerate and has the signature (a, b), then its isotropy index is the minimum of a and b. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space.

Hyperbolic plane

Let F be a field of characteristic not 2 and V = F2. If we consider the general element (x, y) of V, then the quadratic forms q = xy and r = x2y2 are equivalent since there is a linear transformation on V that makes q look like r, and vice versa. Evidently, (V, q) and (V, r) are isotropic. This example is called the hyperbolic plane in the theory of quadratic forms. A common instance has F = real numbers in which case {xV : q(x) = nonzero constant} and {xV : r(x) = nonzero constant} are hyperbolas. In particular, {xV : r(x) = 1} is the unit hyperbola. The notation 1−1 has been used by Milnor and Husemoller [1] :9 for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited.

The affine hyperbolic plane was described by Emil Artin as a quadratic space with basis {M, N} satisfying M2 = N2 = 0, NM = 1, where the products represent the quadratic form. [2]

Through the polarization identity the quadratic form is related to a symmetric bilinear form B(u, v) = 1/4(q(u + v) − q(uv)).

Two vectors u and v are orthogonal when B(u, v) = 0. In the case of the hyperbolic plane, such u and v are hyperbolic-orthogonal.

Split quadratic space

A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension. [1] :57 The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes. [1] :12,3

Relation with classification of quadratic forms

From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field F, classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By Witt's decomposition theorem, every inner product space over a field is an orthogonal direct sum of a split space and an anisotropic space. [1] :56

Field theory

See also

Related Research Articles

Euclidean space Fundamental space of geometry

Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.

Orthogonal group Group of isometries of a Euclidean vector space or, more generally, of a vector space equipped with a quadratic form

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,

Homogeneous space Topological space in group theory

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism (topology). Some authors insist that the action of G be faithful, although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.

In mathematics, a bilinear form on a vector space V is a bilinear map V × VK, where K is the field of scalars. In other words, a bilinear form is a function B : V × VK that is linear in each argument separately:

In mathematics, specifically linear algebra, a degenerate bilinear formf(x, y) on a vector space V is a bilinear form such that the map from V to V given by v is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that

In abstract algebra, a split complex number has two real number components x and y, and is written z = x + yj, where j2 = 1. The conjugate of z is z = xyj. Since j2 = 1, the product of a number z with its conjugate is zz = x2y2, an isotropic quadratic form, N(z) = x2y2.

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.

Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and semisimple algebraic groups are reductive.

Null vector

In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.

In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also referred to more briefly as just symmetric forms when "bilinear" is understood.

In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms over arbitrary fields. The theorem applies to classification of quadratic forms over k and in particular allows one to define the Witt group W(k) which describes the "stable" theory of quadratic forms over the field k.

In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.

In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms

In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form.

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign for every nonzero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite.

In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.

Quadric (algebraic geometry)

In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface


  1. 1 2 3 4 5 Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN   3-540-06009-X. Zbl   0292.10016.
  2. Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive