isotropic quadratic space

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

1. $v\ne 0$ 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: $\ker (Q)=0$; totally isotropic: $\ker (Q)=V$).

Examples.

• Consider the quadratic form $Q(x,y)=x^2+y^2$ in the vector space ${ℝ}^2$ 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 ${ℂ}^2$ over $ℂ$, since $1^2+i^2=0$; the complex numbers are algebraically closed.

• Again, using the same form $x^2+y^2$, but in ${ℝ}^3$ 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.

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