# 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 (or null 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 isotropy index 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.

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 = F2 with elements (x, y). Then the quadratic forms q = xy and r = x2y2 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 {xV : q(x) = nonzero constant} and {xV : r(x) = nonzero constant} are hyperbolas. In particular, {xV : r(x) = 1} is the unit hyperbola. The notation ${\displaystyle \scriptstyle \langle 1\rangle \oplus \langle -1\rangle }$ has been used by Milnor and Huseman[1]:9 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 { M, N } satisfying ${\displaystyle M^{2}=N^{2}=0,\quad NM=1,}$ where the products represent the quadratic form.[2]

Through the polarization identity the quadratic form is related to a symmetric bilinear form ${\displaystyle B(u,v)\ =\ {\frac {1}{4}}(q(u+v)-q(u-v))\ .}$

Two vectors u and v are orthogonal when B(u,v) = 0. 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.[1]:57 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.[1]:12,3

## 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.[1]:56

## 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 Qp of p-adic numbers and (V, q) is a quadratic space of dimension at least five, then it is isotropic.