# 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 *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 *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 **Template:Visible anchor** 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*.

- either

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*.

## Hyperbolic plane

Let *V* = *F*^{2} with elements (*x,y*). Then the quadratic forms *q = xy* and *r* = *x*^{2} − *y*^{2}
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
and
are hyperbolas. In particular, is the unit hyperbola. The notation
has been used by Milnor and Huseman^{[2]} for the hyperbolic plane as the signs of the terms of the bivariate polynomial *r* are exhibited.

## 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]} 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.^{[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.^{[4]}

## Field theory

- 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. - If
*F*is a finite field and (*V*,*q*) is a quadratic space of dimension at least three, then it is isotropic. - 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.

## See also

- Null vector
- Witt group
- Witt ring (forms)
- Witt's theorem
- Symmetric bilinear form
- 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 ISBN 0-8218-1095-2 . - {{#invoke:citation/CS1|citation

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