Inmathematics, given avector spaceX with an associatedquadratic formq, written(X,q), anull vector orisotropic vector is a non-zero elementx ofX for whichq(x) = 0.

In the theory ofrealbilinear forms,definite quadratic forms andisotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

Aquadratic space(X,q) which has a null vector is called apseudo-Euclidean space. The termisotropic vector v whenq(v) = 0 has been used in quadratic spaces,^[1] andanisotropic space for a quadratic space without null vectors.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) intoorthogonal subspacesA andB,X =A +B, whereq is positive-definite onA and negative-definite onB. Thenull cone, orisotropic cone, ofX consists of the union of balanced spheres:$${\displaystyle \bigcup _{r\ge 0}\{x=a+b:q(a)=-q(b)=r,a\in A,b\in B\}.}$$The null cone is also the union of theisotropic lines through the origin.

A composition algebra with a null vector is asplit algebra.^[2]

In acomposition algebra (A, +, ×, *), the quadratic form is q(x) =x x*. Whenx is a null vector then there is no multiplicative inverse forx, and sincex ≠ 0,A is not adivision algebra.

In theCayley–Dickson construction, the split algebras arise in the seriesbicomplex numbers,biquaternions, andbioctonions, which uses thecomplex number field${\displaystyle \mathbb{C}}$ as the foundation of this doubling construction due toL. E. Dickson (1919). In particular, these algebras have twoimaginary units, which commute so their product, when squared, yields +1:

${\displaystyle (hi{)}^2=h^2{i}^2=(-1)(-1)=+1.}$ Then

${\displaystyle (1+hi)(1+hi{)}^{\ast }=(1+hi)(1-hi)=1-(hi{)}^2=0}$ so 1 + hi is a null vector.

The real subalgebras,split complex numbers,split quaternions, andsplit-octonions, with their null cones representing the light tracking into and out of 0 ∈A, suggestspacetime topology.

Thelight-like vectors ofMinkowski space are null vectors.

The fourlinearly independentbiquaternionsl = 1 +hi,n = 1 +hj,m = 1 +hk, andm^∗ = 1 –hk are null vectors and{l,n,m,m^∗ } can serve as abasis for the subspace used to representspacetime. Null vectors are also used in theNewman–Penrose formalism approach to spacetime manifolds.^[3]

In theVerma module of aLie algebra there are null vectors.

• Dubrovin, B. A.;Fomenko, A. T.;Novikov, S. P. (1984).Modern Geometry: Methods and Applications. Translated by Burns, Robert G. Springer. p. 50.ISBN 0-387-90872-2.
• Shaw, Ronald (1982).Linear Algebra and Group Representations. Vol. 1.Academic Press. p. 151.ISBN 0-12-639201-3.
• Neville, E. H. (Eric Harold) (1922).Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions.Cambridge University Press. p. 204.

• Linear algebra
• Quadratic forms

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