In mathematics, a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
over a
field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a
definite quadratic form
In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every non-zero vector of . According to that sign, the quadratic form is called positive-def ...
. More explicitly, if ''q'' is a quadratic form on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or
null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real bilinear forms, definite quadratic forms an ...
) for that quadratic form.
Suppose that is
quadratic space and ''W'' is a
subspace of ''V''. 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 a definite subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.
[
Over the real numbers, more generally in the case where ''F'' is a ]real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
Def ...
(so that the signature is defined), if the quadratic form is non-degenerate and has the signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
, 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 In mathematics and theoretical physics, a pseudo-Euclidean space of signature is a finite- dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vect ...
.
Hyperbolic plane
Let ''F'' be a field of characteristic not 2 and . If we consider the general element of ''V'', then the quadratic forms and are equivalent since there is a linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
on ''V'' that makes ''q'' look like ''r'', and vice versa. Evidently, and are isotropic. This example is called the hyperbolic plane in the theory of quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s. A common instance has ''F'' = real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s in which case and are hyperbola
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
s. In particular, is the unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...
. The notation has been used by Milnor and Husemoller[ 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
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
as a quadratic space with basis satisfying , where the products represent the quadratic form.Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
(1957
''Geometric Algebra'', page 119
via Internet Archive
The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
the quadratic form is related to a symmetric bilinear form
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 biline ...
.
Two vectors ''u'' and ''v'' are orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
when . In the case of the hyperbolic plane, such ''u'' and ''v'' are hyperbolic-orthogonal
In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyp ...
.
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
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^\perp of all vectors in V that are orthogonal to every vector in W. I ...
; equivalently, the index of isotropy is equal to half the dimension.[
]
Relation with classification of quadratic forms
From the point of view of classification of quadratic forms, spaces with definite quadratic forms are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field ''F'', classification of definite quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By Witt's decomposition theorem
:''"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.''
In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any is ...
, every inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
over a field is an orthogonal direct sum
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, m ...
of a split space and a space with definite quadratic form.[
]
Field theory
* If ''F'' is an algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...
field, for example, the field of complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, and is a quadratic space of dimension at least two, then it is isotropic.
* If ''F'' is a finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
and is a quadratic space of dimension at least three, then it is isotropic (this is a consequence of the Chevalley–Warning theorem).
* If ''F'' is the field ''Q''''p'' of ''p''-adic numbers and is a quadratic space of dimension at least five, then it is isotropic.
See also
* Isotropic line
* Polar space
* Witt group
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.
Definition
Fix a field ''k'' of characteristic not equal to 2. All vector spaces w ...
* Witt ring (forms)
* Universal quadratic form
References
* Pete L. Clark
Quadratic forms chapter I: Witts theory
from University of Miami
The University of Miami (UM, UMiami, Miami, U of M, and The U) is a private university, private research university in Coral Gables, Florida, United States. , the university enrolled 19,852 students in two colleges and ten schools across over ...
in Coral Gables, Florida
Coral Gables is a city in Miami-Dade County, Florida, United States. The city is part of the Miami metropolitan area of South Florida and is located southwest of Greater Downtown Miami, Downtown Miami. As of the 2020 United States census, 2020 ...
.
* Tsit Yuen Lam (1973) ''Algebraic Theory of Quadratic Forms'', §1.3 Hyperbolic plane and hyperbolic spaces, W. A. Benjamin.
* Tsit Yuen Lam (2005) ''Introduction to Quadratic Forms over Fields'', American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
.
*
* {{cite book , first=Jean-Pierre , last=Serre , author-link=Jean-Pierre Serre , title=A Course in Arithmetic , volume=7 , publisher=Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, year=2000 , orig-year=1973 , edition=reprint of 3rd , series=Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...
: Classics in mathematics , isbn=0-387-90040-3 , zbl=1034.11003 , url-access=registration , url=https://archive.org/details/courseinarithmet00serr
Quadratic forms
Bilinear forms