Isotropic vector

Definitions

• Is E a vector Space on a body K .

That is to say F a bilinear Form on E . That is to say X a vector of E . It is said that X is an isotropic vector for F if and only if $F\; (X,\; X)\; =0$.

The null vector is always isotropic

• a bilinear form which does not admit that the null vector as isotropic vector is described as anisotropic.

Properties

• When E is a real vector space, a symmetrical bilinear form is anisotropic if and only if it is strictly positive or strictly negative, i.e. if and only if it is a scalar Produit except for the sign.

Indeed if X and is there two vectors such as $F\; (X,\; X)\; >\; 0$ and , one can pose $z=\; ax+y$ and one notes that $F\; (Z,\; Z)\; =\; a^2\; F\; (X,\; X)\; +\; 2a\; F\; (X,\; there)\; +\; F\; (there,\; there)$ is a polynomial of the second degree of negative discriminant. In addition, Z is nonnull because X and is linearly independent there.

• when E is a vector space complexes of size higher or equal to 2, any bilinear form on E admits at least an isotropic vector.

The reasoning above requires only minor adaptations.

• For an antisymmetric bilinear form, any vector is isotropic.

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