isotropic quadratic space
A vector (an element of ) in a quadratic space is isotropic if
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1.
and
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.
Otherwise, it is called anisotropic. A quadratic space is isotropic if it contains an isotropic vector. Otherwise, it is anisotropic. A quadratic space is totally isotropic if every one of its non-zero vector is isotropic, or that .
Similarly, an isotropic quadratic form is one which has a non-trivial kernel, or that there exists a vector such that . 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: ; totally isotropic: ).
Examples.
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Consider the quadratic form in the vector space over the reals. It is clearly anisotropic since there are no real numbers not both , such that .
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However, the same form is isotropic in over , since ; the complex numbers are algebraically closed.
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Again, using the same form , but in over the reals , we see that it is isotropic since the term is missing, so that .
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If we restrict to the subspace consisting of the -axis () and call it , then is totally isotropic, and the -axis is a totally isotropic subspace.
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The quadratic form is clearly isotropic in any vector space over any field. In general, this is true if the coefficients of a diagonal quadratic form consist of ( 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 |