Isotropic Quadratic Form

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 . 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 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 an anisotropic 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.

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. for all non-zero ''v'' in ''V'' ;
:* or ''q'' is ''negative definite'', i.e. for all non-zero ''v'' in ''V''.

More generally, if the quadratic form is non-degenerate and has the signature , 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 . If we consider the general element of ''V'', then the quadratic forms and are equivalent since there is a linear transformation 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 forms. A common instance has ''F'' = real numbers in which case and are hyperbolas. In particular, is the unit hyperbola. 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 as a quadratic space with basis satisfying , where the products represent the quadratic form. Emil Artin (1957
''Geometric Algebra'', page 119
via Internet Archive

Through the polarization identity the quadratic form is related to a symmetric bilinear form .

Two vectors ''u'' and ''v'' are orthogonal when . 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. 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.

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.

Field theory

* If ''F'' is an algebraically closed field, for example, the field of complex numbers, and is a quadratic space of dimension at least two, then it is isotropic.
* If ''F'' is a finite field 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
* Witt ring (forms)
* Universal quadratic form

References

* Pete L. Clark
Quadratic forms chapter I: Witts theory
from University of Miami in Coral Gables, Florida.
* 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 .
*
* {{cite book , first=Jean-Pierre , last=Serre , author-link=Jean-Pierre Serre , title=A Course in Arithmetic , volume=7 , publisher=Springer-Verlag , year=2000 , orig-year=1973 , edition=reprint of 3rd , series=Graduate Texts in Mathematics: Classics in mathematics , isbn=0-387-90040-3 , zbl=1034.11003 , url-access=registration , url=https://archive.org/details/courseinarithmet00serr

Quadratic forms
Bilinear forms