# Isotropic vector

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A non-zero vector that is orthogonal to itself. Let $E$ be a vector space over the field of real or complex numbers and let $\Phi$ be a non-degenerate bilinear form of signature $( p , q )$, $p \neq 0$, $q \neq 0$, on $E \times E$. Then an isotropic vector is a non-zero vector $x \in E$ for which $\Phi ( x , x ) = 0$. One sometimes says that an isotropic vector has zero length (or norm). The set of all isotropic vectors is called the isotropic cone. A subspace $V \subset E$ is called isotropic if there exists a non-zero vector $z \in V$ orthogonal to $V$( that is, the restriction of $\Phi$ to $V \times V$ is degenerate: $V \cap V ^ \perp \neq \{ 0 \}$). A vector subspace $V$ is said to be totally isotropic if all its vectors are isotropic vectors.

In the relativistic interpretation of the Universe, space-time is locally regarded as a four-dimensional vector space with a form of signature $( 3 , 1 )$, the trajectories of photons are isotropic lines, while the isotropic cone is called the light cone.

How to Cite This Entry:
Isotropic vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropic_vector&oldid=47446
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article