isotropic quadratic space

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

  1. 1.

    v0 and

  2. 2.


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: ker(Q)=0; totally isotropic: ker(Q)=V).


  • Consider the quadratic formMathworldPlanetmath Q(x,y)=x2+y2 in the vector spaceMathworldPlanetmath 2 over the reals. It is clearly anisotropic since there are no real numbers a,b not both 0, such that a2+b2=0.

  • However, the same form is isotropic in 2 over , since 12+i2=0; the complex numbersMathworldPlanetmathPlanetmath are algebraically closedMathworldPlanetmath.

  • Again, using the same form x2+y2, but in 3 over the reals , we see that it is isotropic since the z term is missing, so that Q(0,0,1)=02+02=0.

  • If we restrict Q to the subspacePlanetmathPlanetmath consisting of the z-axis (x=y=0) and call it Qz, then Qz is totally isotropic, and the z-axis is a totally isotropic subspace.

  • The quadratic form Q(x,y)=x2-y2 is clearly isotropic in any vector space over any field. In general, this is true if the coefficients of a diagonal quadratic formMathworldPlanetmath Q consist of 1,-1,0 (0 is optional) and nothing else.

Title isotropic quadratic space
Canonical name IsotropicQuadraticSpace
Date of creation 2013-03-22 15:41:57
Last modified on 2013-03-22 15:41:57
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 15A63
Classification msc 11E81
Related topic QuadraticMap2
Related topic QuadraticForm
Defines isotropic vector
Defines isotropic quadratic form
Defines anisotropic vector
Defines anisotropic quadratic form
Defines anisotropic quadratic space
Defines totally isotropic quadratic space
Defines totally isotropic quadratic form