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. 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.
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.
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.
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.
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.