Symplectic vector space

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space subsection</span> </button> <ul id="toc-Standard_symplectic_space-sublist" class="vector-toc-list"> <li id="toc-Lagrangian_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrangian_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Lagrangian form</span> </div> </a> <ul id="toc-Lagrangian_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analogy_with_complex_structures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analogy_with_complex_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Analogy with complex structures</span> </div> </a> <ul id="toc-Analogy_with_complex_structures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Volume_form" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Volume_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Volume form</span> </div> </a> <ul id="toc-Volume_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symplectic_map" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symplectic_map"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Symplectic map</span> </div> </a> <ul id="toc-Symplectic_map-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symplectic_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symplectic_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Symplectic group</span> </div> </a> <ul id="toc-Symplectic_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subspaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subspaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Subspaces</span> </div> </a> <ul id="toc-Subspaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heisenberg_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Heisenberg_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Heisenberg group</span> </div> </a> <ul id="toc-Heisenberg_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div 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href="https://cs.wikipedia.org/wiki/Symplektick%C3%BD_vektorov%C3%BD_prostor" title="Symplektický vektorový prostor – Czech" lang="cs" hreflang="cs" data-title="Symplektický vektorový prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Symplektischer_Vektorraum" title="Symplektischer Vektorraum – German" lang="de" hreflang="de" data-title="Symplektischer Vektorraum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_vectorial_simpl%C3%A9ctico" title="Espacio vectorial simpléctico – Spanish" lang="es" hreflang="es" data-title="Espacio vectorial simpléctico" data-language-autonym="Español" data-language-local-name="Spanish" 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<b>symplectic vector space</b> is a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> (for example the real numbers <span class="mwe-math-element"><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 {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 {R} }"></span>) equipped with a symplectic <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a>. </p><p>A <b>symplectic bilinear form</b> is a <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega :V\times V\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:</mo> <mi>V</mi> <mo>×</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega :V\times V\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d55074a3737e87281a9e6298581b3395b8748641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.153ex; height:2.176ex;" alt="{\displaystyle \omega :V\times V\to F}"></span> that is </p> <dl><dt><a href="/wiki/Bilinear_form" title="Bilinear form">Bilinear</a></dt> <dd><a href="/wiki/Linear_map" title="Linear map">Linear</a> in each argument separately;</dd> <dt><a href="/wiki/Alternating_form" class="mw-redirect" title="Alternating form">Alternating</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (v,v)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (v,v)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9bd569276c2caf23c45ca93083d50ce40b0fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle \omega (v,v)=0}"></span> holds for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}"></span>; and</dd> <dt><a href="/wiki/Nondegenerate_form" class="mw-redirect" title="Nondegenerate form">Non-degenerate</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (v,u)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (v,u)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06fe9853834724c1cbd8d78a45539bbbff37ea83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.007ex; height:2.843ex;" alt="{\displaystyle \omega (v,u)=0}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}"></span> implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c2a269988ef0c55e0449b74950a4976e35a067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle u=0}"></span>.</dd></dl> <p>If the underlying <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> has <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> not 2, alternation is equivalent to <a href="/wiki/Skew_symmetry" class="mw-redirect" title="Skew symmetry">skew-symmetry</a>. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a <a href="/wiki/Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric form</a>, but not vice versa. </p><p>Working in a fixed <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> can be represented by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. The conditions above are equivalent to this matrix being <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a>, <a href="/wiki/Nonsingular_matrix" class="mw-redirect" title="Nonsingular matrix">nonsingular</a>, and <a href="/wiki/Hollow_matrix#Diagonal_entries_all_zero" title="Hollow matrix">hollow</a> (all diagonal entries are zero). This should not be confused with a <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">symplectic matrix</a>, which represents a symplectic transformation of the space. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is <a href="/wiki/Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a>, then its dimension must necessarily be <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even</a> since every skew-symmetric, hollow matrix of odd size has <a href="/wiki/Determinant" title="Determinant">determinant</a> zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Standard_symplectic_space">Standard symplectic space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=1" title="Edit section: Standard symplectic space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Symplectic_matrix#Symplectic_transformations" title="Symplectic matrix">Symplectic matrix § Symplectic transformations</a></div> <p>The standard symplectic space is <span class="mwe-math-element"><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 {R} ^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47460f1a92774729807be11cf62b9178b5771b4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.719ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n}}"></span> with the symplectic form given by a <a href="/wiki/Nonsingular_matrix" class="mw-redirect" title="Nonsingular matrix">nonsingular</a>, <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrix</a>. Typically <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is chosen to be the <a href="/wiki/Block_matrix" title="Block matrix">block matrix</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56f1a2154adeab55502db50289f4f21bc780899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.365ex; height:6.176ex;" alt="{\displaystyle \omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}"></span></dd></dl> <p>where <i>I</i><sub><i>n</i></sub> is the <span class="nowrap"><i>n</i> × <i>n</i></span> <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. In terms of basis vectors <span class="nowrap">(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>, <i>y</i><sub>1</sub>, ..., <i>y<sub>n</sub></i>)</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\omega (x_{i},y_{j})=-\omega (y_{j},x_{i})&=\delta _{ij},\\\omega (x_{i},x_{j})=\omega (y_{i},y_{j})&=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega (x_{i},y_{j})=-\omega (y_{j},x_{i})&=\delta _{ij},\\\omega (x_{i},x_{j})=\omega (y_{i},y_{j})&=0.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2da7feeb60dcf72d7c84530746f16c1b6944ad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.311ex; margin-bottom: -0.194ex; width:28.848ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\omega (x_{i},y_{j})=-\omega (y_{j},x_{i})&=\delta _{ij},\\\omega (x_{i},x_{j})=\omega (y_{i},y_{j})&=0.\end{aligned}}}"></span></dd></dl> <p>A modified version of the <a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a> shows that any finite-dimensional symplectic vector space has a basis such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> takes this form, often called a <i><b>Darboux basis</b></i> or <a href="/wiki/Symplectic_basis" title="Symplectic basis">symplectic basis</a>. </p><p><b>Sketch of process:</b> </p><p>Start with an arbitrary basis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},...,v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1},...,v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed315d96db6db7b0f8d128d1347e287935132df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.698ex; height:2.009ex;" alt="{\displaystyle v_{1},...,v_{n}}"></span>, and represent the dual of each basis vector by the <a href="/wiki/Dual_basis" title="Dual basis">dual basis</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (v_{i},\cdot )=\sum _{j}\omega (v_{i},v_{j})v_{j}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (v_{i},\cdot )=\sum _{j}\omega (v_{i},v_{j})v_{j}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89928e7985d79f2371938e3bba4efd8ef35e644c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.139ex; height:5.843ex;" alt="{\displaystyle \omega (v_{i},\cdot )=\sum _{j}\omega (v_{i},v_{j})v_{j}^{*}}"></span>. This gives us a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matrix with entries <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (v_{i},v_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (v_{i},v_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83bffa0ca04c02ae6fe5c8890c08375c6237165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.254ex; height:3.009ex;" alt="{\displaystyle \omega (v_{i},v_{j})}"></span>. Solve for its null space. Now for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{1},...,\lambda _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{1},...,\lambda _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465e61de40dd57b954d544cd16248adb4f97a90d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.962ex; height:2.843ex;" alt="{\displaystyle (\lambda _{1},...,\lambda _{n})}"></span> in the null space, we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\omega (v_{i},\cdot )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\omega (v_{i},\cdot )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d96adf125ca60f14b92bece49626abc738d424c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.866ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\omega (v_{i},\cdot )=0}"></span>, so the null space gives us the degenerate subspace <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae15ff9b845587dc4e1816f59c3fed0e71a132f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{0}}"></span>. </p><p>Now arbitrarily pick a complementary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V_{0}\oplus W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊕</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V_{0}\oplus W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a33721db00362c36dc6a98ac456857b6078e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.571ex; height:2.509ex;" alt="{\displaystyle V=V_{0}\oplus W}"></span>, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1},...,w_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1},...,w_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7911b0b3957596e5065ce98c2c414c723df884e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.227ex; height:2.009ex;" alt="{\displaystyle w_{1},...,w_{m}}"></span> be a basis of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},\cdot )\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},\cdot )\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86703656dfa0e2433007c45124e22b9d0fbdceb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.915ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},\cdot )\neq 0}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},w_{1})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},w_{1})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a726b61e226a5f7ac05875a575fce1ba8948379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},w_{1})=0}"></span>, WLOG <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},w_{2})\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},w_{2})\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a66b010c2ad7c560e52cdb15d8ea192f8c42ef32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},w_{2})\neq 0}"></span>. Now scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8998e0957bb573a19e7d9d934ced62ee68ab8fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle w_{2}}"></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},w_{2})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},w_{2})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70eeb7c5f0e4d6a0cf02310a0079a78c82f45567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},w_{2})=1}"></span>. Then define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>w</mi> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9995ace37e68b7ecfe179c4514ff9fafb9bef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.572ex; height:3.009ex;" alt="{\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}}"></span> for each of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=w_{3},w_{4},...,w_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=w_{3},w_{4},...,w_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b490fddca9f3e0e85a7bd56d572d5cb12b574f4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.742ex; height:2.009ex;" alt="{\displaystyle w=w_{3},w_{4},...,w_{m}}"></span>. Iterate. </p><p>Notice that this method applies for symplectic vector space over any field, not just the field of real numbers. </p><p><b>Case of real or complex field:</b> </p><p>When the space is over the field of real numbers, then we can modify the modified Gram-Schmidt process as follows: Start the same way. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1},...,w_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1},...,w_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7911b0b3957596e5065ce98c2c414c723df884e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.227ex; height:2.009ex;" alt="{\displaystyle w_{1},...,w_{m}}"></span> be an orthonormal basis (with respect to the usual inner product on <span class="mwe-math-element"><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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},\cdot )\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},\cdot )\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86703656dfa0e2433007c45124e22b9d0fbdceb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.915ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},\cdot )\neq 0}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},w_{1})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},w_{1})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a726b61e226a5f7ac05875a575fce1ba8948379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},w_{1})=0}"></span>, WLOG <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},w_{2})\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},w_{2})\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a66b010c2ad7c560e52cdb15d8ea192f8c42ef32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},w_{2})\neq 0}"></span>. Now multiply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8998e0957bb573a19e7d9d934ced62ee68ab8fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle w_{2}}"></span> by a sign, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (w_{1},w_{2})\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (w_{1},w_{2})\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4aae265bab9b44095f4797535ee7dcc9690af40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle \omega (w_{1},w_{2})\geq 0}"></span>. Then define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>w</mi> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9995ace37e68b7ecfe179c4514ff9fafb9bef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.572ex; height:3.009ex;" alt="{\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}}"></span> for each of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=w_{3},w_{4},...,w_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=w_{3},w_{4},...,w_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b490fddca9f3e0e85a7bd56d572d5cb12b574f4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.742ex; height:2.009ex;" alt="{\displaystyle w=w_{3},w_{4},...,w_{m}}"></span>, then scale each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98af407af5c02e29010c7563af95f8986026679c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.349ex; height:2.509ex;" alt="{\displaystyle w'}"></span> so that it has norm one. Iterate. </p><p>Similarly, for the field of complex numbers, we may choose a unitary basis. This proves the <a href="/wiki/Skew-symmetric_matrix#Spectral_theory" title="Skew-symmetric matrix">spectral theory of antisymmetric matrices</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Lagrangian_form">Lagrangian form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=2" title="Edit section: Lagrangian form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is another way to interpret this standard symplectic form. Since the model space <b>R</b><sup>2<i>n</i></sup> used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let <i>V</i> be a real vector space of dimension <i>n</i> and <i>V</i><sup>∗</sup> its <a href="/wiki/Dual_space" title="Dual space">dual space</a>. Now consider the <a href="/wiki/Direct_sum_of_vector_spaces" class="mw-redirect" title="Direct sum of vector spaces">direct sum</a> <span class="nowrap"><i>W</i> = <i>V</i> ⊕ <i>V</i><sup>∗</sup></span> of these spaces equipped with the following form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (x\oplus \eta ,y\oplus \xi )=\xi (x)-\eta (y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊕</mo> <mi>η</mi> <mo>,</mo> <mi>y</mi> <mo>⊕</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (x\oplus \eta ,y\oplus \xi )=\xi (x)-\eta (y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc43db563ab2bfe7b46998044f819eaf5014a452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.543ex; height:2.843ex;" alt="{\displaystyle \omega (x\oplus \eta ,y\oplus \xi )=\xi (x)-\eta (y).}"></span></dd></dl> <p>Now choose any <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> <span class="nowrap">(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i></sub>)</span> of <i>V</i> and consider its <a href="/wiki/Dual_space" title="Dual space">dual basis</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(v_{1}^{*},\ldots ,v_{n}^{*}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(v_{1}^{*},\ldots ,v_{n}^{*}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4af76eb21242ff915815a7ee73fdaba3fb2fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.549ex; height:3.009ex;" alt="{\displaystyle \left(v_{1}^{*},\ldots ,v_{n}^{*}\right).}"></span></dd></dl> <p>We can interpret the basis vectors as lying in <i>W</i> if we write <span class="nowrap"><i>x</i><sub><i>i</i></sub> = (<i>v</i><sub><i>i</i></sub>, 0) and <i>y</i><sub><i>i</i></sub> = (0, <i>v</i><sub><i>i</i></sub><sup>∗</sup>)</span>. Taken together, these form a complete basis of <i>W</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef16837575746c7e03165c01ef8a9bc6f7f65da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.33ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}).}"></span></dd></dl> <p>The form <i>ω</i> defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form <span class="nowrap"><i>V</i> ⊕ <i>V</i><sup>∗</sup></span>. The subspace <i>V</i> is not unique, and a choice of subspace <i>V</i> is called a <b>polarization</b>. The subspaces that give such an isomorphism are called <b>Lagrangian subspaces</b> or simply <b>Lagrangians</b>. </p><p>Explicitly, given a Lagrangian subspace <a href="#Subspaces">as defined below</a>, then a choice of basis <span class="nowrap">(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>)</span> defines a dual basis for a complement, by <span class="nowrap"><i>ω</i>(<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>j</i></sub>) = <i>δ</i><sub><i>ij</i></sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Analogy_with_complex_structures">Analogy with complex structures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=3" title="Edit section: Analogy with complex structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Just as every symplectic structure is isomorphic to one of the form <span class="nowrap"><i>V</i> ⊕ <i>V</i><sup>∗</sup></span>, every <a href="/wiki/Linear_complex_structure" title="Linear complex structure"><i>complex</i> structure</a> on a vector space is isomorphic to one of the form <span class="nowrap"><i>V</i> ⊕ <i>V</i></span>. Using these structures, the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> of an <i>n</i>-manifold, considered as a 2<i>n</i>-manifold, has an <a href="/wiki/Almost_complex_structure" class="mw-redirect" title="Almost complex structure">almost complex structure</a>, and the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle"><i>co</i>tangent bundle</a> of an <i>n</i>-manifold, considered as a 2<i>n</i>-manifold, has a symplectic structure: <span class="nowrap"><i>T</i><sub>∗</sub>(<i>T</i><sup>∗</sup><i>M</i>)<sub><i>p</i></sub> = <i>T</i><sub><i>p</i></sub>(<i>M</i>) ⊕ (<i>T</i><sub><i>p</i></sub>(<i>M</i>))<sup>∗</sup></span>. </p><p>The complex analog to a Lagrangian subspace is a <a href="/wiki/Real_subspace" class="mw-redirect" title="Real subspace"><i>real</i> subspace</a>, a subspace whose <a href="/wiki/Complexification" title="Complexification">complexification</a> is the whole space: <span class="nowrap"><i>W</i> = <i>V</i> ⊕ <i>J</i> <i>V</i></span>. As can be seen from the standard symplectic form above, every symplectic form on <b>R</b><sup>2<i>n</i></sup> is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on <b>C</b><sup><i>n</i></sup> (with the convention of the first argument being anti-linear). </p> <div class="mw-heading mw-heading2"><h2 id="Volume_form">Volume form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=4" title="Edit section: Volume form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>ω</i> be an <a href="/wiki/Alternating_bilinear_form" class="mw-redirect" title="Alternating bilinear form">alternating bilinear form</a> on an <i>n</i>-dimensional real vector space <i>V</i>, <span class="nowrap"><i>ω</i> ∈ Λ<sup>2</sup>(<i>V</i>)</span>. Then <i>ω</i> is non-degenerate if and only if <i>n</i> is even and <span class="nowrap"><i>ω</i><sup><i>n</i>/2</sup> = <i>ω</i> ∧ ... ∧ <i>ω</i></span> is a <a href="/wiki/Volume_form" title="Volume form">volume form</a>. A volume form on a <i>n</i>-dimensional vector space <i>V</i> is a non-zero multiple of the <i>n</i>-form <span class="nowrap"><i>e</i><sub>1</sub><sup>∗</sup> ∧ ... ∧ <i>e</i><sub><i>n</i></sub><sup>∗</sup></span> where <span class="nowrap"><i>e</i><sub>1</sub>, <i>e</i><sub>2</sub>, ..., <i>e</i><sub><i>n</i></sub></span> is a basis of <i>V</i>. </p><p>For the standard basis defined in the previous section, we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{n}=(-1)^{\frac {n}{2}}x_{1}^{*}\wedge \dotsb \wedge x_{n}^{*}\wedge y_{1}^{*}\wedge \dotsb \wedge y_{n}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>∧</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{n}=(-1)^{\frac {n}{2}}x_{1}^{*}\wedge \dotsb \wedge x_{n}^{*}\wedge y_{1}^{*}\wedge \dotsb \wedge y_{n}^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd09c6d000433d8953c15deffc6a6ee2f5d6215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.922ex; height:3.843ex;" alt="{\displaystyle \omega ^{n}=(-1)^{\frac {n}{2}}x_{1}^{*}\wedge \dotsb \wedge x_{n}^{*}\wedge y_{1}^{*}\wedge \dotsb \wedge y_{n}^{*}.}"></span></dd></dl> <p>By reordering, one can write </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{n}=x_{1}^{*}\wedge y_{1}^{*}\wedge \dotsb \wedge x_{n}^{*}\wedge y_{n}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>∧</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>∧</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{n}=x_{1}^{*}\wedge y_{1}^{*}\wedge \dotsb \wedge x_{n}^{*}\wedge y_{n}^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac3321e7f70eb7da745b6b6c6eb9a98fd668a8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.968ex; height:3.009ex;" alt="{\displaystyle \omega ^{n}=x_{1}^{*}\wedge y_{1}^{*}\wedge \dotsb \wedge x_{n}^{*}\wedge y_{n}^{*}.}"></span></dd></dl> <p>Authors variously define <i>ω</i><sup><i>n</i></sup> or (−1)<sup><i>n</i>/2</sup><i>ω</i><sup><i>n</i></sup> as the <b>standard volume form</b>. An occasional factor of <i>n</i>! may also appear, depending on whether the definition of the <a href="/wiki/Alternating_product" class="mw-redirect" title="Alternating product">alternating product</a> contains a factor of <i>n</i>! or not. The volume form defines an <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a> on the symplectic vector space <span class="nowrap">(<i>V</i>, <i>ω</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Symplectic_map">Symplectic map</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=5" title="Edit section: Symplectic map"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <span class="nowrap">(<i>V</i>, <i>ω</i>)</span> and <span class="nowrap">(<i>W</i>, <i>ρ</i>)</span> are symplectic vector spaces. Then a <a href="/wiki/Linear_map" title="Linear map">linear map</a> <span class="nowrap"><i>f</i> : <i>V</i> → <i>W</i></span> is called a <b>symplectic map</b> if the <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a> preserves the symplectic form, i.e. <span class="nowrap"><i>f</i><span style="padding-left:0.12em;"><sup>∗</sup></span><i>ρ</i> = <i>ω</i></span>, where the pullback form is defined by <span class="nowrap">(<i>f</i><span style="padding-left:0.12em;"><sup>∗</sup></span><i>ρ</i>)(<i>u</i>, <i>v</i>) = <i>ρ</i>(<i>f</i>(<i>u</i>), <i>f</i>(<i>v</i>))</span>. Symplectic maps are volume- and orientation-preserving. </p> <div class="mw-heading mw-heading2"><h2 id="Symplectic_group">Symplectic group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=6" title="Edit section: Symplectic group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="nowrap"><i>V</i> = <i>W</i></span>, then a symplectic map is called a <b>linear symplectic transformation</b> of <i>V</i>. In particular, in this case one has that <span class="nowrap"><i>ω</i>(<i>f</i>(<i>u</i>), <i>f</i>(<i>v</i>)) = <i>ω</i>(<i>u</i>, <i>v</i>)</span>, and so the <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> <i>f</i> preserves the symplectic form. The set of all symplectic transformations forms a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and in particular a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, called the <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic group</a> and denoted by Sp(<i>V</i>) or sometimes <span class="nowrap">Sp(<i>V</i>, <i>ω</i>)</span>. In matrix form symplectic transformations are given by <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">symplectic matrices</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Subspaces">Subspaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=7" title="Edit section: Subspaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>W</i> be a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <i>V</i>. Define the <b>symplectic complement</b> of <i>W</i> to be the subspace </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{\perp }=\{v\in V\mid \omega (v,w)=0{\mbox{ for all }}w\in W\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>∣</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for all </mtext> </mstyle> </mrow> <mi>w</mi> <mo>∈</mo> <mi>W</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{\perp }=\{v\in V\mid \omega (v,w)=0{\mbox{ for all }}w\in W\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f58b214c3193a98e1bf43b13c6cd2b322f22c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.047ex; height:3.176ex;" alt="{\displaystyle W^{\perp }=\{v\in V\mid \omega (v,w)=0{\mbox{ for all }}w\in W\}.}"></span></dd></dl> <p>The symplectic complement satisfies: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(W^{\perp }\right)^{\perp }&=W\\\dim W+\dim W^{\perp }&=\dim V.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>W</mi> </mtd> </mtr> <mtr> <mtd> <mi>dim</mi> <mo>⁡</mo> <mi>W</mi> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mi>V</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(W^{\perp }\right)^{\perp }&=W\\\dim W+\dim W^{\perp }&=\dim V.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c093aa39a1c285b3293e60710107636a00e172d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.365ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\left(W^{\perp }\right)^{\perp }&=W\\\dim W+\dim W^{\perp }&=\dim V.\end{aligned}}}"></span></dd></dl> <p>However, unlike <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">orthogonal complements</a>, <i>W</i><sup>⊥</sup> ∩ <i>W</i> need not be 0. We distinguish four cases: </p> <ul><li><i>W</i> is <b>symplectic</b> if <span class="nowrap"><i>W</i><sup>⊥</sup> ∩ <i>W</i> = {0</span>}. This is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>ω</i> restricts to a nondegenerate form on <i>W</i>. A symplectic subspace with the restricted form is a symplectic vector space in its own right.</li> <li><i>W</i> is <b>isotropic</b> if <span class="nowrap"><i>W</i> ⊆ <i>W</i><sup>⊥</sup></span>. This is true if and only if <i>ω</i> restricts to 0 on <i>W</i>. Any one-dimensional subspace is isotropic.</li> <li><i>W</i> is <b>coisotropic</b> if <span class="nowrap"><i>W</i><sup>⊥</sup> ⊆ <i>W</i></span>. <i>W</i> is coisotropic if and only if <i>ω</i> descends to a nondegenerate form on the <a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a> <i>W</i>/<i>W</i><sup>⊥</sup>. Equivalently <i>W</i> is coisotropic if and only if <i>W</i><sup>⊥</sup> is isotropic. Any <a href="/wiki/Codimension" title="Codimension">codimension</a>-one subspace is coisotropic.</li> <li><i>W</i> is <b>Lagrangian</b> if <span class="nowrap"><i>W</i> = <i>W</i><sup>⊥</sup></span>. A subspace is Lagrangian if and only if it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of <i>V</i>. Every isotropic subspace can be extended to a Lagrangian one.</li></ul> <p>Referring to the canonical vector space <b>R</b><sup>2<i>n</i></sup> above, </p> <ul><li>the subspace spanned by {<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>} is symplectic</li> <li>the subspace spanned by {<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>} is isotropic</li> <li>the subspace spanned by {<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub>, <i>y</i><sub>1</sub>} is coisotropic</li> <li>the subspace spanned by {<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub>} is Lagrangian.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Heisenberg_group">Heisenberg group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=8" title="Edit section: Heisenberg group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a></div> <p>A <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a> can be defined for any symplectic vector space, and this is the typical way that <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg groups</a> arise. </p><p>A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>, meaning with trivial Lie bracket. The Heisenberg group is a <a href="/wiki/Central_extension_(mathematics)" class="mw-redirect" title="Central extension (mathematics)">central extension</a> of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relations</a> (CCR), and a Darboux basis corresponds to <a href="/wiki/Canonical_coordinate" class="mw-redirect" title="Canonical coordinate">canonical coordinates</a> – in physics terms, to <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operators</a> and <a href="/wiki/Position_operator" title="Position operator">position operators</a>. </p><p>Indeed, by the <a href="/wiki/Stone%E2%80%93von_Neumann_theorem" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a>, every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one. </p><p>Further, the <a href="/wiki/Group_ring" title="Group ring">group algebra</a> of (the dual to) a vector space is the <a href="/wiki/Symmetric_algebra" title="Symmetric algebra">symmetric algebra</a>, and the group algebra of the Heisenberg group (of the dual) is the <a href="/wiki/Weyl_algebra" title="Weyl algebra">Weyl algebra</a>: one can think of the central extension as corresponding to quantization or <a href="/wiki/Deformation_quantization" title="Deformation quantization">deformation</a>. </p><p>Formally, the symmetric algebra of a vector space <i>V</i> over a field <i>F</i> is the group algebra of the dual, <span class="nowrap">Sym(<i>V</i>) := <i>F</i>[<i>V</i><sup>∗</sup>]</span>, and the Weyl algebra is the group algebra of the (dual) Heisenberg group <span class="nowrap"><i>W</i>(<i>V</i>) = <i>F</i>[<i>H</i>(<i>V</i><sup>∗</sup>)]</span>. Since passing to group algebras is a <a href="/wiki/Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant functor</a>, the central extension map <span class="nowrap"><i>H</i>(<i>V</i>) → <i>V</i></span> becomes an inclusion <span class="nowrap">Sym(<i>V</i>) → <i>W</i>(<i>V</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> is a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> with a smoothly-varying <i>closed</i> symplectic form on each <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a>.</li> <li><a href="/wiki/Maslov_index" class="mw-redirect" title="Maslov index">Maslov index</a></li> <li>A <a href="/wiki/Symplectic_representation" title="Symplectic representation">symplectic representation</a> is a <a href="/wiki/Group_representation" title="Group representation">group representation</a> where each group element acts as a symplectic transformation.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symplectic_vector_space&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Claude_Godbillon&action=edit&redlink=1" class="new" title="Claude Godbillon (page does not exist)">Claude Godbillon</a> (1969) "Géométrie différentielle et mécanique analytique", Hermann</li> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAbrahamMarsden1978" class="citation book cs1"><a href="/wiki/Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Abraham, Ralph</a>; <a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden, Jerrold E.</a> (1978). "Hamiltonian and Lagrangian Systems". <i>Foundations of Mechanics</i> (2nd ed.). London: Benjamin-Cummings. pp. 161–252. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-0102-X" title="Special:BookSources/0-8053-0102-X"><bdi>0-8053-0102-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hamiltonian+and+Lagrangian+Systems&rft.btitle=Foundations+of+Mechanics&rft.place=London&rft.pages=161-252&rft.edition=2nd&rft.pub=Benjamin-Cummings&rft.date=1978&rft.isbn=0-8053-0102-X&rft.aulast=Abraham&rft.aufirst=Ralph&rft.au=Marsden%2C+Jerrold+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymplectic+vector+space" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/25029/1/FoM2.pdf">PDF</a></li> <li>Paulette Libermann and Charles-Michel Marle (1987) "Symplectic Geometry and Analytical Mechanics", D. Reidel</li> <li>Jean-Marie Souriau (1997) "Structure of Dynamical Systems, A Symplectic View of Physics", Springer</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Symplectic_vector_space&oldid=1240257215">https://en.wikipedia.org/w/index.php?title=Symplectic_vector_space&oldid=1240257215</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Symplectic_geometry" title="Category:Symplectic geometry">Symplectic geometry</a></li><li><a href="/wiki/Category:Bilinear_forms" title="Category:Bilinear forms">Bilinear forms</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 August 2024, at 11:50<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Symplectic vector space - Wikipedia