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{{Short description|Vector on which a quadratic form is zero}}
{{about|zeros of a quadratic form|the zero element in a vector space|Zero vector|null vectors in Minkowski space|Minkowski space#Causal structure}}
[[File:Conformalsphere.pdf|thumb|A null cone where <math>q(x,y,z) = x^2 + y^2 - z^2 .</math>]]

In [[mathematics]], given a [[vector space]] ''X'' with an associated [[quadratic form]] ''q'', written {{nowrap|(''X'', ''q'')}}, a '''null vector''' or '''isotropic vector''' is a non-zero element ''x'' of ''X'' for which {{nowrap|1=''q''(''x'') = 0}}.

In the theory of [[real number|real]] [[bilinear form]]s, [[definite quadratic form]]s and [[isotropic quadratic form]]s are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

A [[quadratic space]] {{nowrap|(''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,<ref>[[Emil Artin]] (1957) [[Geometric Algebra (book)|''Geometric Algebra'']], [https://archive.org/details/geometricalgebra033556mbp/page/n129/mode/2up?view=theater&q=isotropic isotropic]</ref> and '''anisotropic space''' for a quadratic space without null vectors.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into [[orthogonal subspaces]] ''A'' and ''B'', {{nowrap|1=''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:
<math display="block">\bigcup_{r \geq 0} \{x = a + b : q(a) = -q(b) = r, \ \ a \in A, b \in B \}.</math>
The null cone is also the union of the [[isotropic line]]s through the origin.

==Split algebras==
A composition algebra with a null vector is a '''split algebra'''.<ref>Arthur A. Sagle & Ralph E. Walde (1973) ''Introduction to Lie Groups and Lie Algebras'', page 197, [[Academic Press]]</ref>

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]].

In the [[Cayley–Dickson construction]], the split algebras arise in the series [[bicomplex number]]s, [[biquaternion]]s, and [[bioctonion]]s, which uses the [[complex number]] field <math>\Complex</math> as the foundation of this doubling construction due to [[L. E. Dickson]] (1919). In particular, these algebras have two [[imaginary unit]]s, which commute so their product, when squared, yields +1:
:<math>(hi)^2 = h^2 i^2 = (-1)(-1) = +1 .</math> Then
:<math>(1 + hi)(1 + hi)^* = (1 +hi)(1 - hi) = 1 - (hi)^2 = 0</math> so 1 + hi is a null vector.
The real subalgebras, [[split complex number]]s, [[split quaternion]]s, and [[split-octonion]]s, with their null cones representing the light tracking into and out of 0 ∈ ''A'', suggest [[spacetime topology]].

==Examples==
The [[Minkowski space#Causal structure|light-like]] vectors of [[Minkowski space]] are null vectors.

The four [[linearly independent]] [[biquaternion]]s {{nowrap|1=''l'' = 1 + ''hi''}}, {{nowrap|1=''n'' = 1 + ''hj''}}, {{nowrap|1=''m'' = 1 + ''hk''}}, and {{nowrap|1=''m''<sup>∗</sup> = 1 – ''hk''}} are null vectors and {{nowrap|{ ''l'', ''n'', ''m'', ''m''<sup>∗</sup> }{{void}}}} can serve as a [[basis (linear algebra)|basis]] for the subspace used to represent [[spacetime]]. Null vectors are also used in the [[Newman–Penrose formalism]] approach to spacetime manifolds.<ref>Patrick Dolan (1968) [http://projecteuclid.org/euclid.cmp/1103840725 A Singularity-free solution of the Maxwell-Einstein Equations], [[Communications in Mathematical Physics]] 9(2):161–8, especially 166, link from [[Project Euclid]]</ref>

In the [[Verma module]] of a [[Lie algebra]] there are null vectors.

==References==
{{Reflist}}

* {{cite book |first=B. A. |last=Dubrovin |first2=A. T. |last2=Fomenko |authorlink2=Anatoly Fomenko |first3=S. P. |last3=Novikov |authorlink3=Sergei Novikov (mathematician) |year=1984 |title=Modern Geometry: Methods and Applications |translator-first=Robert G. |translator-last=Burns |page=[https://archive.org/details/moderngeometryme000dubr/page/50 50] |publisher=Springer |isbn=0-387-90872-2 |url=https://archive.org/details/moderngeometryme000dubr |url-access=registration }}
* {{cite book |first=Ronald |last=Shaw |year=1982 |title=Linear Algebra and Group Representations |volume=1 |page=151 |publisher=[[Academic Press]] |isbn=0-12-639201-3 |url=https://books.google.com/books?id=C6DgAAAAMAAJ }}
* {{cite book | last = Neville | first = E. H. (Eric Harold) | author-link =Eric Harold Neville | title =Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions | publisher =[[Cambridge University Press]] | date = 1922 |page=[https://archive.org/details/prolegomenatoana00nevi/page/204 204]| url =https://archive.org/details/prolegomenatoana00nevi }}

[[Category:Linear algebra]]
[[Category:Quadratic forms]]

Null vector - Revision history

~2025-31953-87 at 15:08, 7 November 2025