This article is about zeros of a quadratic form. For the zero element in a vector space, see Zero vector. For null vectors in Minkowski space, see Minkowski space §Causal structure.
In mathematics, given a vector spaceX with an associated quadratic formq, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.
A quadratic space(X, q) which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces,[1] and anisotropic space for a quadratic space without null vectors.
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspacesA and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres:
The null cone is also the union of the isotropic lines through the origin.
Split algebras
A composition algebra with a null vector is a split algebra.[2]
In a composition algebra (A, +, ×, *), the quadratic form is q(x) = x x*. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a division algebra.
This article uses material from the Wikipedia article Isotropic_vector, and is written by contributors.
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