In mathematics, aquadratic form over afieldF is said to beisotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is adefinite quadratic form. More explicitly, ifq is a quadratic form on avector spaceV overF, then a non-zero vectorv inV is said to beisotropic ifq(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (ornull vector) for that quadratic form.
Suppose that(V,q) isquadratic space andW is asubspace ofV. ThenW is called anisotropic subspace ofV ifsome vector in it is isotropic, atotally isotropic subspace ifall vectors in it are isotropic, and adefinite subspace if it does not containany (non-zero) isotropic vectors. Theisotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.[1]
Over the real numbers, more generally in the case whereF is areal closed field (so that the signature is defined), if the quadratic form is non-degenerate and has thesignature(a,b), then its isotropy index is the minimum ofa andb. An important example of an isotropic form over the reals occurs inpseudo-Euclidean space.
LetF be a field ofcharacteristic not 2 andV =F2. If we consider the general element(x,y) ofV, then the quadratic formsq =xy andr =x2 −y2 are equivalent since there is alinear transformation onV that makesq look liker, and vice versa. Evidently,(V,q) and(V,r) are isotropic. This example is called thehyperbolic plane in the theory ofquadratic forms. A common instance hasF =real numbers in which case{x ∈V :q(x) = nonzero constant} and{x ∈V :r(x) = nonzero constant} arehyperbolas. In particular,{x ∈V :r(x) = 1} is theunit hyperbola. The notation⟨1⟩ ⊕ ⟨−1⟩ has been used by Milnor and Husemoller[1]: 9 for the hyperbolic plane as the signs of the terms of thebivariate polynomialr are exhibited.
The affine hyperbolic plane was described byEmil Artin as a quadratic space with basis{M,N} satisfyingM2 =N2 = 0,NM = 1, where the products represent the quadratic form.[2]
Through thepolarization identity the quadratic form is related to asymmetric bilinear formB(u,v) =1/4(q(u +v) −q(u −v)).
Two vectorsu andv areorthogonal whenB(u,v) = 0. In the case of the hyperbolic plane, suchu andv arehyperbolic-orthogonal.
A space with quadratic form issplit (ormetabolic) if there is a subspace which is equal to its ownorthogonal complement; equivalently, the index of isotropy is equal to half the dimension.[1]: 57 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.[1]: 12, 3
From the point of view of classification of quadratic forms, spaces with definite quadratic forms are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general fieldF, classification of definite quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. ByWitt's decomposition theorem, everyinner product space over a field is anorthogonal direct sum of a split space and a space with definite quadratic form.[1]: 56