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. 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.

Hyperbolic plane

Not to be confused with the plane in hyperbolic geometry.

Let V = F² with elements (x, y). 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 $\scriptstyle \langle 1\rangle \oplus \langle -1\rangle$ has been used by Milnor and Huseman for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited.

Split quadratic space

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. 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.

Relation with classification of quadratic forms

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.

Field theory

• 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 Q_p of p-adic numbers and (V, q) is a quadratic space of dimension at least five, then it is isotropic.

See also

• Polar space
• Witt group
• Witt ring (forms)
• Witt's theorem
• Symmetric bilinear form
• Universal quadratic form

References

• Pete L. Clark, Quadratic forms chapter I: Witts theory from University of Miami in Coral Gables, Florida.
• Tsit Yuen Lam (1973) Algebraic Theory of Quadratic Forms, §1.3 Hyperbolic plane and hyperbolic spaces, W. A. Benjamin.
• Tsit Yuen Lam (2005) Introduction to Quadratic Forms over Fields, American Mathematical Society ISBN 0-8218-1095-2 .
• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
• O'Meara, O.T (1963). Introduction to Quadratic Forms. Springer-Verlag. p. 94 §42D Isotropy. ISBN 3-540-66564-1. {{cite book}}: ISBN / Date incompatibility (help)
• Serre, Jean-Pierre (2000) [1973]. A Course in Arithmetic. Graduate Texts in Mathematics: Classics in mathematics. Vol. 7 (reprint of 3rd ed.). Springer-Verlag. ISBN 0-387-90040-3. Zbl 1034.11003.