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Concept
Isotropic quadratic form
Summary
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 explicitly, 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 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.
Let F be a field of characteristic not 2 and V = F2. If we consider the general element (x, y) of V, then the quadratic forms q = xy and r = x2 − y2 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 {x ∈ V : q(x) = nonzero constant} and {x ∈ V : r(x) = nonzero constant} are hyperbolas. In particular, {x ∈ V : r(x) = 1} is the unit hyperbola. The notation 1 ⊕ −1 has been used by Milnor and Husemoller for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited.
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