Isotropic quadratic form

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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 for that quadratic form.

Suppose that (V, q) 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 (non-zero) isotropic vectors. The isotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.[1]

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.

Hyperbolic plane[edit]

Not to be confused with Hyperbolic plane in geometry.

Let V = F2 with elements (x, y). Then the quadratic forms q = xy and r = x2y2 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 {xV : q(x) = nonzero constant} and {xV : r(x) = nonzero constant} are hyperbolas. In particular, {xV : r(x) = 1} is the unit hyperbola. The notation \scriptstyle \langle 1 \rangle \oplus \langle -1 \rangle has been used by Milnor and Huseman[2] for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited.

Split quadratic space[edit]

A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement: equivalently, the index of isotropy is equal to half the dimension.[1] 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.[3]

Relation with classification of quadratic forms[edit]

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.[4]

Field theory[edit]

  • If F is an algebraically closed field, for example, the field of complex numbers, and (V, q) is a quadratic space of dimension at least two, then it is isotropic.
  • If F is a finite field and (V, q) is a quadratic space of dimension at least three, then it is isotropic.
  • If F is the field Qp of p-adic numbers and (V, q) is a quadratic space of dimension at least five, then it is isotropic.

See also[edit]

References[edit]

  1. ^ a b Milnor & Husemoller (1973) p.57
  2. ^ Milnor & Husemoller (1973) page 9
  3. ^ Milnor & Husemoller (1973) pp.12–13
  4. ^ Milnor & Husemoller (1973) p.56