Isotropic quadratic form

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, it is anisotropic. More explicitly, 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 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 of V. 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.

Over the real numbers, more generally in the case where F is a real closed field (so that the signature is defined), if the quadratic form is non-degenerate and has the signature(a, b), then its isotropy index is the minimum of a and b. An anisotropic quadratic form over a real closed field is always definite. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space.