In mathematics, a quadratic form over a fieldF 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 spaceV over F, then a non-zero vector v in V is said to be isotropic if. A quadratic form is isotropic if and only ifthere exists a non-zero isotropic 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 isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. A quadratic form q on a finite-dimensional real vector spaceV is anisotropic if and only if q is a definite form: More generally, if the quadratic form is non-degenerate and has the signature, then its isotropy index is the minimum of a and b.
Hyperbolic plane
Let F be a field of characteristic not 2 and. If we consider the general element ofV, 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 unithyperbola. The notation has been used by Milnor and Huseman for the hyperbolic plane as the signs of the terms of the bivariate polynomialr are exhibited. The affine hyperbolic plane was described by Emil Artin as a quadratic space with basis satisfying, where the products represent the quadratic form. Through the polarization identity the quadratic form is related to a symmetric bilinear form. Two vectors u and v are orthogonal when. In the case of the hyperbolic plane, such u and v are hyperbolic-orthogonal.
Split quadratic space
A space with quadratic form is split 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.
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.
