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<title>Null vector</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Null vector</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about zeros of a quadratic form. For the zero element in a vector space, see <a href="Zero_vector" class="mw-redirect" title="Zero vector">Zero vector</a>. For null vectors in Minkowski space, see <a href="Minkowski_space#Causal_structure" title="Minkowski space">Minkowski space § Causal structure</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, given a <a href="Vector_space" title="Vector space">vector space</a> <i>X</i> with an associated <a href="Quadratic_form" title="Quadratic form">quadratic form</a> <i>q</i>, written <span class="nowrap">(<i>X</i>, <i>q</i>)</span>, a <b>null vector</b> or <b>isotropic vector</b> is a non-zero element <i>x</i> of <i>X</i> for which <span class="nowrap"><i>q</i>(<i>x</i>) = 0</span>.

</p><p>In the theory of <a href="Real_number" title="Real number">real</a> <a href="Bilinear_form" title="Bilinear form">bilinear forms</a>, <a href="Definite_quadratic_form" title="Definite quadratic form">definite quadratic forms</a> and <a href="Isotropic_quadratic_form" title="Isotropic quadratic form">isotropic quadratic forms</a> are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

</p><p>A <a href="Quadratic_space" class="mw-redirect" title="Quadratic space">quadratic space</a> <span class="nowrap">(<i>X</i>, <i>q</i>)</span> which has a null vector is called a <a href="Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a>. The term <i>isotropic vector v</i> when <i>q</i>(<i>v</i>) = 0 has been used in quadratic spaces,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <b>anisotropic space</b> for a quadratic space without null vectors.

</p><p>A pseudo-Euclidean vector space may be decomposed (non-uniquely) into <a href="Orthogonal_subspaces" class="mw-redirect" title="Orthogonal subspaces">orthogonal subspaces</a> <i>A</i> and <i>B</i>, <span class="nowrap"><i>X</i> = <i>A</i> + <i>B</i></span>, where <i>q</i> is positive-definite on <i>A</i> and negative-definite on <i>B</i>. The <b>null cone</b>, or <b>isotropic cone</b>, of <i>X</i> consists of the union of balanced spheres:

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<annotation encoding="application/x-tex">{\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.}</annotation>

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The null cone is also the union of the <a href="Isotropic_line" title="Isotropic line">isotropic lines</a> through the origin.

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<div class="mw-heading mw-heading2"><h2 id="Split_algebras">Split algebras</h2></div>

<p>A composition algebra with a null vector is a <b>split algebra</b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>In a <a href="Composition_algebra" title="Composition algebra">composition algebra</a> (<i>A</i>, +, ×, *), the quadratic form is q(<i>x</i>) = <i>x x</i>*. When <i>x</i> is a null vector then there is no multiplicative inverse for <i>x</i>, and since <i>x</i> ≠ 0, <i>A</i> is not a <a href="Division_algebra" title="Division algebra">division algebra</a>.

</p><p>In the <a href="Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a>, the split algebras arise in the series <a href="Bicomplex_number" title="Bicomplex number">bicomplex numbers</a>, <a href="Biquaternion" title="Biquaternion">biquaternions</a>, and <a href="Bioctonion" title="Bioctonion">bioctonions</a>, which uses the <a href="Complex_number" title="Complex number">complex number</a> field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span> as the foundation of this doubling construction due to <a href="L._E._Dickson" class="mw-redirect" title="L. E. Dickson">L. E. Dickson</a> (1919). In particular, these algebras have two <a href="Imaginary_unit" title="Imaginary unit">imaginary units</a>, which commute so their product, when squared, yields +1:

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<annotation encoding="application/x-tex">{\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}</annotation>

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</math></span><img src="./a972dae66835f1ddbfb0709775f2598527695a48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.727ex; height:3.176ex;" alt="{\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}" loading="lazy"></span> Then</dd>

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<annotation encoding="application/x-tex">{\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}</annotation>

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</math></span><img src="./a56f59d11c9204244cae2a762297111450041603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.334ex; height:3.176ex;" alt="{\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}" loading="lazy"></span> so 1 + hi is a null vector.</dd></dl>

<p>The real subalgebras, <a href="Split_complex_number" class="mw-redirect" title="Split complex number">split complex numbers</a>, <a href="Split_quaternion" class="mw-redirect" title="Split quaternion">split quaternions</a>, and <a href="Split-octonion" title="Split-octonion">split-octonions</a>, with their null cones representing the light tracking into and out of 0 ∈ <i>A</i>, suggest <a href="Spacetime_topology" title="Spacetime topology">spacetime topology</a>.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>The <a href="Minkowski_space#Causal_structure" title="Minkowski space">light-like</a> vectors of <a href="Minkowski_space" title="Minkowski space">Minkowski space</a> are null vectors.

</p><p>The four <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> <a href="Biquaternion" title="Biquaternion">biquaternions</a> <span class="nowrap"><i>l</i> = 1 + <i>hi</i></span>, <span class="nowrap"><i>n</i> = 1 + <i>hj</i></span>, <span class="nowrap"><i>m</i> = 1 + <i>hk</i></span>, and <span class="nowrap"><i>m</i><sup>∗</sup> = 1 – <i>hk</i></span> are null vectors and <span class="nowrap">{ <i>l</i>, <i>n</i>, <i>m</i>, <i>m</i><sup>∗</sup> }</span> can serve as a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for the subspace used to represent <a href="Spacetime" title="Spacetime">spacetime</a>. Null vectors are also used in the <a href="Newman%E2%80%93Penrose_formalism" title="Newman–Penrose formalism">Newman–Penrose formalism</a> approach to spacetime manifolds.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>In the <a href="Verma_module" title="Verma module">Verma module</a> of a <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> there are null vectors.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Emil_Artin" title="Emil Artin">Emil Artin</a> (1957) <a href="Geometric_Algebra_(book)" title="Geometric Algebra (book)"><i>Geometric Algebra</i></a>, <a rel="nofollow" class="external text" href="https://archive.org/details/geometricalgebra033556mbp/page/n129/mode/2up?view=theater&q=isotropic">isotropic</a></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Arthur A. Sagle & Ralph E. Walde (1973) <i>Introduction to Lie Groups and Lie Algebras</i>, page 197, <a href="Academic_Press" title="Academic Press">Academic Press</a></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Patrick Dolan (1968) <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1103840725">A Singularity-free solution of the Maxwell-Einstein Equations</a>, <a href="Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a> 9(2):161–8, especially 166, link from <a href="Project_Euclid" title="Project Euclid">Project Euclid</a></span>

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</style><cite id="CITEREFDubrovinFomenkoNovikov1984" class="citation book cs1">Dubrovin, B. A.; <a href="Anatoly_Fomenko" title="Anatoly Fomenko">Fomenko, A. T.</a>; <a href="Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov, S. P.</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/moderngeometryme000dubr"><i>Modern Geometry: Methods and Applications</i></a></span>. Translated by Burns, Robert G. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/moderngeometryme000dubr/page/50">50</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90872-2</bdi>.</cite></li>

<li><cite id="CITEREFShaw1982" class="citation book cs1">Shaw, Ronald (1982). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=C6DgAAAAMAAJ"><i>Linear Algebra and Group Representations</i></a>. Vol. 1. <a href="Academic_Press" title="Academic Press">Academic Press</a>. p. 151. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-639201-3</bdi>.</cite></li>

<li><cite id="CITEREFNeville1922" class="citation book cs1"><a href="Eric_Harold_Neville" title="Eric Harold Neville">Neville, E. H. (Eric Harold)</a> (1922). <a rel="nofollow" class="external text" href="https://archive.org/details/prolegomenatoana00nevi"><i>Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions</i></a>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/prolegomenatoana00nevi/page/204">204</a>.</cite></li></ul></div><div><div class="zim-footer">

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