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isotropic quadratic space (Definition)

A vector $ v$ (an element of $ V$) in a quadratic space $ (V,Q)$ is isotropic if

  1. $ v\ne0$ and
  2. $ Q(v)=0$.
Otherwise, it is called anisotropic. A quadratic space $ (V,Q)$ is isotropic if it contains an isotropic vector. Otherwise, it is anisotropic. A quadratic space $ (V,Q)$ is totally isotropic if every one of its non-zero vector is isotropic, or that $ Q(V)=0$.

Similarly, an isotropic quadratic form is one which has a non-trivial kernel, or that there exists a vector $ v$ such that $ Q(v)=0$. The definitions for that of an anisotropic quadratic form and that of a totally isotropic quadratic form should now be clear from the above discussion (anisotropic: $ \operatorname{ker}(Q)=0$; totally isotropic: $ \operatorname{ker}(Q)=V$).

Examples.

  • Consider the quadratic form $ Q(x,y)=x^2+y^2$ in the vector space $ \mathbb{R}^2$ over the reals. It is clearly anisotropic since there are no real numbers $ a,b$ not both 0, such that $ a^2+b^2=0$.
  • However, the same form is isotropic in $ \mathbb{C}^2$ over $ \mathbb{C}$, since $ 1^2+i^2=0$; the complex numbers are algebraically closed.
  • Again, using the same form $ x^2+y^2$, but in $ \mathbb{R}^3$ over the reals , we see that it is isotropic since the $ z$ term is missing, so that $ Q(0,0,1)=0^2+0^2=0$.
  • If we restrict $ Q$ to the subspace consisting of the $ z$-axis ($ x=y=0$) and call it $ Q_z$, then $ Q_z$ is totally isotropic, and the $ z$-axis is a totally isotropic subspace.
  • The quadratic form $ Q(x,y)=x^2-y^2$ is clearly isotropic in any vector space over any field. In general, this is true if the coefficients of a diagonal quadratic form $ Q$ consist of $ 1, -1, 0$ (0 is optional) and nothing else.



"isotropic quadratic space" is owned by CWoo.
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See Also: quadratic map, quadratic form

Also defines:  isotropic vector, isotropic quadratic form, anisotropic vector, anisotropic quadratic form, anisotropic quadratic space, totally isotropic quadratic space, totally isotropic quadratic form
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Cross-references: diagonal quadratic form, coefficients, field, subspace, term, algebraically closed, complex numbers, reals, vector space, quadratic form, clear, definitions, kernel, non-zero vector, totally isotropic, contains, quadratic space, vector
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This is version 7 of isotropic quadratic space, born on 2006-02-20, modified 2006-12-14.
Object id is 7643, canonical name is IsotropicQuadraticSpace.
Accessed 4432 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
 11E81 (Number theory :: Forms and linear algebraic groups :: Algebraic theory of quadratic forms; Witt groups and rings)
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