Isotropic quadratic form


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 it 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 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 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 isotropic subspaces.

Examples

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 ("p","q"), then its isotropy index is the minimum of "p" and "q".

2. If "F" is an algebraically closed field, for example, the field of complex numbers,and ("V","q") is a quadratic space of dimension at least two, then it is isotropic. 3. If "F" is a finite field and ("V","q") is a quadratic space of dimension at least three, then it is isotropic.

4. If "F" is the field "Q""p" of p-adic numbers and ("V","q") is a quadratic space of dimension at least five, then it is isotropic.

5. A hyperbolic plane is a two-dimensional quadratic space with form "x"2-"y"2.

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.

See also

*Null vector
*Witt group
*Symmetric bilinear form

References

* Serre, Jean-Pierre, "A course in arithmetic". Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.
* Milnor, John and Dale Husemoller, "Symmetric bilinear forms". Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. 1973.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Quadratic form — In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including… …   Wikipedia

  • Degenerate form — For other uses, see Degeneracy. In mathematics, specifically linear algebra, a degenerate bilinear form ƒ(x,y) on a vector space V is one such that the map from V to V * (the dual space of V) given by is not an isomorphism. An equivalent… …   Wikipedia

  • Isotropy — is uniformity in all orientations; it is derived from the Greek iso (equal) and tropos (direction). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy.… …   Wikipedia

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Hyperbolic plane — In mathematics, the term hyperbolic plane may refer to:* A two dimensional quadratic space with a non singular isotropic quadratic form * A plane in hyperbolic geometry …   Wikipedia

  • Spinor — In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the… …   Wikipedia

  • Spin representation — In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are… …   Wikipedia

  • Hasse–Minkowski theorem — In mathematics, the Hasse–Minkowski theorem states that a quadratic form is isotropic globally if and only if it is everywhere isotropic locally; it is the classic local global principle. Here to be isotropic means to that there is some non zero… …   Wikipedia

  • Symplectic manifold — In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2 form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology.… …   Wikipedia

  • Spinors in three dimensions — In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group… …   Wikipedia