Bilinear forms | Quadratic forms

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 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. 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. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space. (Wikipedia).

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Witt group | Linear subspace | Vector space | Universal quadratic form | Finite field | Chevalley–Warning theorem | Polynomial | Emil Artin | Hyperbolic geometry | Polarization identity | Pseudo-Euclidean space | Null vector | Polar space | Symmetric bilinear form | Field (mathematics) | Real number | Isotropic line | Hyperbola | Quadratic form | P-adic number | Inner product space | Plane (geometry) | Unit hyperbola | Orthogonal complement