isotropic quadratic space
A vector v (an element of V) in a quadratic space (V,Q) is isotropic if
-
1.
v≠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.
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•
Consider the quadratic form
Q(x,y)=x2+y2 in the vector space
ℝ2 over the reals. It is clearly anisotropic since there are no real numbers a,b not both 0, such that a2+b2=0.
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•
However, the same form is isotropic in ℂ2 over ℂ, since 12+i2=0; the complex numbers
are algebraically closed
.
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•
Again, using the same form x2+y2, but in ℝ3 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 subspace
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 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 |