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.


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