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<title>Drehmatrix</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Drehmatrix</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Eine Drehmatrix oder Rotationsmatrix ist eine <a href="Reelle_Zahl" title="Reelle Zahl">reelle</a>, <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrix</a> mit <a href="Determinante" title="Determinante">Determinante</a> +1. Ihre <a href="Matrix-Vektor-Produkt" title="Matrix-Vektor-Produkt">Multiplikation</a> mit einem <a href="Vektor" title="Vektor">Vektor</a> lässt sich interpretieren als (sogenannte aktive) <a href="Drehung" title="Drehung">Drehung</a> des Vektors im <a href="Euklidischer_Vektorraum" class="mw-redirect" title="Euklidischer Vektorraum">euklidischen Raum</a> oder als passive Drehung des <a href="Koordinatensystem" title="Koordinatensystem">Koordinatensystems</a>, dann mit umgekehrtem Drehsinn. Bei der passiven Drehung ändert sich der Vektor nicht, er hat bloß je eine Darstellung (Koordinatenwerte) im alten und im neuen Koordinatensystem. Dabei handelt es sich stets um Drehungen um den <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Ursprung</a>, da die Multiplikation einer Matrix mit dem <a href="Nullvektor" title="Nullvektor">Nullvektor</a> diesen auf sich selbst <a href="Lineare_Abbildung" title="Lineare Abbildung">abbildet</a>.

In ungeraden <a href="Dimension_(Mathematik)#Dimension_eines_Vektorraumes_(Hamel-Dimension)" title="Dimension (Mathematik)">Dimensionen</a> werden durch eine Drehung weitere Vektoren auf sich selbst abgebildet, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rp=p}">

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<annotation encoding="application/x-tex">{\displaystyle Rp=p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4aa37a2437212a7ecb9726802555f1781100e4c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.201ex; height:2.509ex;" alt="{\displaystyle Rp=p}" loading="lazy">. Im dreidimensionalen Raum handelt es sich also um eine Gerade, die Drehachse. Eine Drehmatrix enthält <a href="Trigonometrische_Funktion" title="Trigonometrische Funktion">trigonometrische</a> Ausdrücke des Drehwinkels und der Orientierung des invarianten Unterraumes. In geraden Dimensionen muss die Drehmatrix keinen reellen <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> haben.

<div class="mw-heading mw-heading2"><h2 id="Drehmatrix_der_Ebene_ℝ²">Drehmatrix der Ebene ℝ²</h2></div>

In der <a href="Euklidische_Ebene" class="mw-redirect" title="Euklidische Ebene">euklidischen Ebene</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"> wird die Drehung eines Vektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> (aktive Drehung, Überführung in den Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'}">

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<annotation encoding="application/x-tex">{\displaystyle p'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40e623e3163571a220ed60ecb31aa78c24104b85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}" loading="lazy">) um einen festen <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Ursprung</a> um den <a href="Winkel" title="Winkel">Winkel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> im mathematisch positiven Sinn (gegen den <a href="Uhrzeigersinn" class="mw-redirect" title="Uhrzeigersinn">Uhrzeigersinn</a>) durch die Multiplikation mit der Drehmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle R_{\alpha }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97d43bbd9121a279f3f818b4d2d5b7d52985456d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.048ex; height:2.509ex;" alt="{\displaystyle R_{\alpha }}" loading="lazy"> erreicht:

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<annotation encoding="application/x-tex">{\displaystyle p'=R_{\alpha }p\qquad (1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92cf2857ed121bd1768a6fef1ae193bd65b65f00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.877ex; height:3.009ex;" alt="{\displaystyle p'=R_{\alpha }p\qquad (1)}" loading="lazy"></dd></dl>

Jede Rotation um den Ursprung ist eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a>. Wie bei jeder linearen Abbildung genügt daher zur Festlegung der Gesamtabbildung die Festlegung der Bilder der Elemente einer beliebigen <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</a>. Wird die <a href="Standardbasis" title="Standardbasis">Standardbasis</a> gewählt, sind die Bilder der Basisvektoren gerade die Spalten der dazugehörigen <a href="Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a>.

Hier wirkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\end{pmatrix}}\mapsto {\begin{pmatrix}\cos \alpha \\\sin \alpha \end{pmatrix}}\quad }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87dcf65367813454d51d2ddf6bbde839f4ca651a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.43ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\end{pmatrix}}\mapsto {\begin{pmatrix}\cos \alpha \\\sin \alpha \end{pmatrix}}\quad }" loading="lazy">und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\begin{pmatrix}0\\1\end{pmatrix}}\mapsto {\begin{pmatrix}-\sin \alpha \\\cos \alpha \end{pmatrix}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \quad {\begin{pmatrix}0\\1\end{pmatrix}}\mapsto {\begin{pmatrix}-\sin \alpha \\\cos \alpha \end{pmatrix}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9957f29644b4f47d729fc7cef8b986c1d7271b24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.017ex; height:6.176ex;" alt="{\displaystyle \quad {\begin{pmatrix}0\\1\end{pmatrix}}\mapsto {\begin{pmatrix}-\sin \alpha \\\cos \alpha \end{pmatrix}}.}" loading="lazy"></dd></dl>

Für die Drehmatrix einer Drehung um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle R_{\alpha }={\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a0d6b0ea73b388fead8071b239b83d597335b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.2ex; height:6.176ex;" alt="{\displaystyle R_{\alpha }={\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}.}" loading="lazy"></dd></dl>

Ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle P=(x,y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/099c643e67f1973c546d7e001eae4ab0ec7b2415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.172ex; height:2.843ex;" alt="{\displaystyle P=(x,y)}" loading="lazy"> geht in den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'=(x',y')}">

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<annotation encoding="application/x-tex">{\displaystyle P'=(x',y')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99abc598efef2eed0d4ffb370d102153e8e8d551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.308ex; height:3.009ex;" alt="{\displaystyle P'=(x',y')}" loading="lazy"> über, dessen (als Spaltenvektor geschriebenen) Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'={\begin{pmatrix}x'\\y'\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle p'={\begin{pmatrix}x'\\y'\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d36df6a00c90ae88d29c079aa19fec3de8673a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:11.229ex; height:6.176ex;" alt="{\displaystyle p'={\begin{pmatrix}x'\\y'\end{pmatrix}}}" loading="lazy"> man aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\begin{pmatrix}x\\y\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle p={\begin{pmatrix}x\\y\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01285318e9c5b6be804a7bb0a0931e0c7cb895dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:9.86ex; height:6.176ex;" alt="{\displaystyle p={\begin{pmatrix}x\\y\end{pmatrix}}}" loading="lazy"> durch Anwenden der obigen Formel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}">

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<annotation encoding="application/x-tex">{\displaystyle (1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a25115739469707c4758b189fe310a750092a80a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}" loading="lazy"> erhält:

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<annotation encoding="application/x-tex">{\displaystyle p'=R_{\alpha }\cdot p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69310de88bea4dbf2b295df232990a8252d53ed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.939ex; height:2.843ex;" alt="{\displaystyle p'=R_{\alpha }\cdot p}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}={\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}\cdot {\begin{pmatrix}x\\y\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/802b4bbc638a48833cdde9503566a33091ed90b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.874ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}={\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}\cdot {\begin{pmatrix}x\\y\end{pmatrix}}}" loading="lazy"></dd></dl>

Diese <a href="Matrixmultiplikation" class="mw-redirect" title="Matrixmultiplikation">Matrixmultiplikation</a> ergibt:

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<annotation encoding="application/x-tex">{\displaystyle x'=x\cdot \cos \alpha -y\cdot \sin \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d0c80ce53a8b2ead570ca0bad3ba84a64b32cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.513ex; height:2.843ex;" alt="{\displaystyle x'=x\cdot \cos \alpha -y\cdot \sin \alpha }" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y'=x\cdot \sin \alpha +y\cdot \cos \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27e692eaf396599aa5a8ab60072c2ed9b278edd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.344ex; height:2.843ex;" alt="{\displaystyle y'=x\cdot \sin \alpha +y\cdot \cos \alpha }" loading="lazy"></dd></dl>

Bei der passiven Drehung wird das Koordinatensystem mathematisch positiv gedreht. Der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> erscheint im gedrehten Koordinatensystem als im Uhrzeigersinn zurückgedrehter Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy">. Seine Koordinaten im gedrehten Koordinatensystem findet man durch Multiplikation mit der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\alpha }^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{\alpha }^{-1}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}=R_{\alpha }^{-1}p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00f50a0ff0609281198c3720420bfc1928c2127d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.814ex; height:3.009ex;" alt="{\displaystyle {\hat {p}}=R_{\alpha }^{-1}p}" loading="lazy"></dd></dl>

Die Drehmatrix für die passive Drehung ist:

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<annotation encoding="application/x-tex">{\displaystyle R_{\alpha }^{-1}={\begin{pmatrix}\cos \alpha &\sin \alpha \\-\sin \alpha &\cos \alpha \end{pmatrix}}=R_{-\alpha }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2df27d40a8619779def576369622afb418242d00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.027ex; height:6.176ex;" alt="{\displaystyle R_{\alpha }^{-1}={\begin{pmatrix}\cos \alpha &\sin \alpha \\-\sin \alpha &\cos \alpha \end{pmatrix}}=R_{-\alpha }}" loading="lazy"></dd></dl>

Die Verkettung zweier positiver Drehungen um die Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> ist erneut eine Drehung, und zwar um den Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha +\beta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99b80a3fdecb9cf75091789bb4335a1bb3561b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha +\beta }" loading="lazy"> (siehe auch <a href="Kreisgruppe" title="Kreisgruppe">Kreisgruppe</a>).

Die zur Verkettung gehörende Matrix kann mittels <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Multiplikation</a> aus den beiden einzelnen Drehmatrizen berechnet werden:

<mtr>

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<mtr>

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<mtd>

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{\alpha +\beta }=&R_{\alpha }R_{\beta }\\{\begin{pmatrix}\cos(\alpha +\beta )&-\sin(\alpha +\beta )\\\sin(\alpha +\beta )&\cos(\alpha +\beta )\end{pmatrix}}=&{\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}{\begin{pmatrix}\cos \beta &-\sin \beta \\\sin \beta &\cos \beta \end{pmatrix}}\\=&{\begin{pmatrix}\cos \alpha \cos \beta -\sin \alpha \sin \beta &-\cos \alpha \sin \beta -\sin \alpha \cos \beta \\\cos \alpha \sin \beta +\sin \alpha \cos \beta &\cos \alpha \cos \beta -\sin \alpha \sin \beta \end{pmatrix}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfb7c57a6af6d9abc801910a0a120998d45042ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:86.962ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}R_{\alpha +\beta }=&R_{\alpha }R_{\beta }\\{\begin{pmatrix}\cos(\alpha +\beta )&-\sin(\alpha +\beta )\\\sin(\alpha +\beta )&\cos(\alpha +\beta )\end{pmatrix}}=&{\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}{\begin{pmatrix}\cos \beta &-\sin \beta \\\sin \beta &\cos \beta \end{pmatrix}}\\=&{\begin{pmatrix}\cos \alpha \cos \beta -\sin \alpha \sin \beta &-\cos \alpha \sin \beta -\sin \alpha \cos \beta \\\cos \alpha \sin \beta +\sin \alpha \cos \beta &\cos \alpha \cos \beta -\sin \alpha \sin \beta \end{pmatrix}}\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Drehmatrizen_des_Raumes_ℝ³"> Drehmatrizen des Raumes ℝ³</h2></div>

Die elementaren Drehungen im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> sind Drehungen um die üblichen <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatenachsen</a>. Die folgenden Matrizen drehen einen Punkt (bzw. Vektor) um den <a href="Winkel" title="Winkel">Winkel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> bei festen Koordinatenachsen. In der Physik werden häufig Drehungen des Koordinatensystems benutzt, dann müssen bei den untenstehenden Matrizen die <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> aller Sinus-Einträge geändert werden. Die Drehung eines Vektors um einen bestimmten Winkel in einem Koordinatensystem führt auf dieselben Spaltenvektoren wie die Drehung des Koordinatensystems um den gleichen Winkel in umgekehrter Richtung (Drehung um negativen Winkel).

Die Matrizen gelten sowohl für <a href="Rechtssystem_(Mathematik)" title="Rechtssystem (Mathematik)">Rechts-</a> als auch für Linkssysteme. Drehungen mit positiven Drehwinkeln sind im Rechtssystem Drehungen entgegen dem Uhrzeigersinn. Im Linkssystem wird bei positiven Winkeln mit dem Uhrzeigersinn gedreht. Der Drehsinn ergibt sich, wenn man entgegen der positiven Drehachse auf den Ursprung schaut. In Rechtssystemen kann auch eine <a href="Drei-Finger-Regel" title="Drei-Finger-Regel">Rechte-Hand-Regel</a> angewandt werden: Zeigt der Daumen der rechten Hand in Richtung der Drehachse, so geben die gebeugten restlichen Finger die Richtung des Drehwinkels an. Im Ergebnis ist das Vorzeichen der Sinus-Einträge der Drehung um die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

<ul><li>Drehung um die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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<annotation encoding="application/x-tex">{\displaystyle R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70c39b1c4aba6804b353d7e214c945707941698b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.869ex; height:9.176ex;" alt="{\displaystyle R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}}}" loading="lazy"></dd></dl>

</mstyle>

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

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c78bc461e6843e2ef823fe93266d966514948f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.746ex; height:9.176ex;" alt="{\displaystyle R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}}}" loading="lazy"></dd></dl>

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-Achse:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39ecb068d85b21a9429e63f794256927d2107ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.698ex; height:9.176ex;" alt="{\displaystyle R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}" loading="lazy"></dd></dl>

<ul><li>Drehung um eine <a href="Ursprungsgerade" title="Ursprungsgerade">Ursprungsgerade</a>, deren Richtung und Orientierung durch den beliebigen <a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}=(n_{1},n_{2},n_{3})^{\mathsf {T}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}=(n_{1},n_{2},n_{3})^{\mathsf {T}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5d317a5f2b6dbdd0d1f5f6e349de472e92ae743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.069ex; height:3.176ex;" alt="{\displaystyle {\hat {n}}=(n_{1},n_{2},n_{3})^{\mathsf {T}}}" loading="lazy"> gegeben ist:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle R_{\hat {n}}(\alpha )={\begin{pmatrix}n_{1}^{2}\left(1-\cos \alpha \right)+\cos \alpha &n_{1}n_{2}\left(1-\cos \alpha \right)-n_{3}\sin \alpha &n_{1}n_{3}\left(1-\cos \alpha \right)+n_{2}\sin \alpha \\n_{2}n_{1}\left(1-\cos \alpha \right)+n_{3}\sin \alpha &n_{2}^{2}\left(1-\cos \alpha \right)+\cos \alpha &n_{2}n_{3}\left(1-\cos \alpha \right)-n_{1}\sin \alpha \\n_{3}n_{1}\left(1-\cos \alpha \right)-n_{2}\sin \alpha &n_{3}n_{2}\left(1-\cos \alpha \right)+n_{1}\sin \alpha &n_{3}^{2}\left(1-\cos \alpha \right)+\cos \alpha \end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c53456d03dbe08f894ef7626684e40ee3d61ab46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:98.311ex; height:10.176ex;" alt="{\displaystyle R_{\hat {n}}(\alpha )={\begin{pmatrix}n_{1}^{2}\left(1-\cos \alpha \right)+\cos \alpha &n_{1}n_{2}\left(1-\cos \alpha \right)-n_{3}\sin \alpha &n_{1}n_{3}\left(1-\cos \alpha \right)+n_{2}\sin \alpha \\n_{2}n_{1}\left(1-\cos \alpha \right)+n_{3}\sin \alpha &n_{2}^{2}\left(1-\cos \alpha \right)+\cos \alpha &n_{2}n_{3}\left(1-\cos \alpha \right)-n_{1}\sin \alpha \\n_{3}n_{1}\left(1-\cos \alpha \right)-n_{2}\sin \alpha &n_{3}n_{2}\left(1-\cos \alpha \right)+n_{1}\sin \alpha &n_{3}^{2}\left(1-\cos \alpha \right)+\cos \alpha \end{pmatrix}}}" loading="lazy"></dd></dl>

Diese beliebige Drehung lässt sich auch über drei aufeinanderfolgende Drehungen mit den <a href="Eulersche_Winkel" title="Eulersche Winkel">eulerschen Winkeln</a> um bestimmte Koordinatenachsen erzielen, sodass sich diese Matrix auch mit diesen Winkeln formulieren lässt.

Eine Drehung um eine beliebige Achse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}\cdot {\hat {n}}=1}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}\cdot {\hat {n}}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/528684240153aee128d29177deed8ed522bf63aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.729ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}\cdot {\hat {n}}=1}" loading="lazy">) um den Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle R_{\hat {n}}(\alpha ){\vec {x}}={\hat {n}}({\hat {n}}\cdot {\vec {x}})+\cos \left(\alpha \right)({\hat {n}}\times {\vec {x}})\times {\hat {n}}+\sin \left(\alpha \right)({\hat {n}}\times {\vec {x}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60ef97b864c2343bc66f8d31b51d91f226ea0125.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.54ex; height:3.009ex;" alt="{\displaystyle R_{\hat {n}}(\alpha ){\vec {x}}={\hat {n}}({\hat {n}}\cdot {\vec {x}})+\cos \left(\alpha \right)({\hat {n}}\times {\vec {x}})\times {\hat {n}}+\sin \left(\alpha \right)({\hat {n}}\times {\vec {x}})}" loading="lazy"></dd></dl>

Dies lässt sich mit der <a href="Kreuzprodukt#Graßmann-Identität" title="Kreuzprodukt">Graßmann-Identität</a> für doppelte Kreuzprodukte und dem <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadischen Produkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"> umformen zu:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{\hat {n}}(\alpha ){\vec {x}}&=(1-\cos \alpha ){\hat {n}}({\hat {n}}\cdot {\vec {x}})+\cos \alpha \,{\vec {x}}+\sin \alpha ({\hat {n}}\times {\vec {x}})\\&={\Big (}(1-\cos \alpha ){\hat {n}}\otimes {\hat {n}}+I\,\cos \alpha +\sin \alpha \sum _{i}({\hat {n}}\times {\hat {e}}_{i})\otimes {\hat {e}}_{i}{\Big )}{\vec {x}}\\&={\Big (}(1-\cos \alpha ){\hat {n}}\otimes {\hat {n}}+I\,\cos \alpha +[{\hat {n}}]_{\times }\,\sin \alpha {\Big )}{\vec {x}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85501f010db1469432007d48a280f4b72b6b0687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:66.144ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}R_{\hat {n}}(\alpha ){\vec {x}}&=(1-\cos \alpha ){\hat {n}}({\hat {n}}\cdot {\vec {x}})+\cos \alpha \,{\vec {x}}+\sin \alpha ({\hat {n}}\times {\vec {x}})\\&={\Big (}(1-\cos \alpha ){\hat {n}}\otimes {\hat {n}}+I\,\cos \alpha +\sin \alpha \sum _{i}({\hat {n}}\times {\hat {e}}_{i})\otimes {\hat {e}}_{i}{\Big )}{\vec {x}}\\&={\Big (}(1-\cos \alpha ){\hat {n}}\otimes {\hat {n}}+I\,\cos \alpha +[{\hat {n}}]_{\times }\,\sin \alpha {\Big )}{\vec {x}}\end{aligned}}}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{i}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/accb899e7a6343250cfd70556c42482bb10d30fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.509ex;" alt="{\displaystyle {\hat {e}}_{i}}" loading="lazy"> sind die kanonischen Einheitsvektoren. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\hat {n}}]_{\times }}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [{\hat {n}}]_{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17ce7758304520d1967285d880d50bd371766ee7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.199ex; height:2.843ex;" alt="{\displaystyle [{\hat {n}}]_{\times }}" loading="lazy"> ist die <a href="Kreuzprodukt#Kreuzproduktmatrix" title="Kreuzprodukt">Kreuzproduktmatrix</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

</semantics>

Der Term in geschweiften Klammern stellt die Drehmatrix im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

<msub>

<mover>

</mover>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [R_{\hat {n}}(\alpha )]_{ij}=(1-\cos \alpha )n_{i}n_{j}+\cos \alpha \,\delta _{ij}+\sin \alpha \,\varepsilon _{ikj}n_{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbeb1bf06c1e055d542f6de59ea56e2856d69bac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.027ex; height:3.009ex;" alt="{\displaystyle [R_{\hat {n}}(\alpha )]_{ij}=(1-\cos \alpha )n_{i}n_{j}+\cos \alpha \,\delta _{ij}+\sin \alpha \,\varepsilon _{ikj}n_{k}}" loading="lazy"></dd></dl>

Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"> das <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ikj}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ikj}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98e065e80ff22aa49b8cb27ba1a14f52eff36e09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.417ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{ikj}}" loading="lazy"> das <a href="Levi-Civita-Symbol" title="Levi-Civita-Symbol">Levi-Civita-Symbol</a>.

Eine Drehmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\neq I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\neq I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1670e55a3e3d1f2ad972f098ee10a8281c51b5e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.034ex; height:2.676ex;" alt="{\displaystyle R\neq I}" loading="lazy"> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> hat den <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> 1, dieser ist nicht entartet, und der zugehörige <a href="Eigenwertproblem#Eigenraum_zum_Eigenwert" class="mw-redirect" title="Eigenwertproblem">Eigenraum</a> bildet die Drehachse.

<div class="mw-heading mw-heading3"><h3 id="Parametrisierung">Parametrisierung</h3></div>

Für Drehmatrizen im dreidimensionalen Raum sind mehrere Parametrisierungen bekannt:

<ul><li><a href="Eulersche_Winkel" title="Eulersche Winkel">Euler-, Kardan- und Tait-Bryan-Winkel</a> werden in der <a href="Kreiseltheorie" title="Kreiseltheorie">Kreiseltheorie</a>, <a href="Luftfahrt" title="Luftfahrt">Luftfahrt</a>, <a href="Schifffahrt" title="Schifffahrt">Schifffahrt</a> und im <a href="Automobilbau" class="mw-redirect" title="Automobilbau">Automobilbau</a> verwendet.</li>

<li>Die <a href="Euler-Rodrigues-Formel" title="Euler-Rodrigues-Formel">Euler-Rodrigues-Formel</a> basiert auf den <a href="Quaternion" title="Quaternion">Quaternionen</a> und wird in der <a href="Robotik" title="Robotik">Robotik</a> und <a href="Computergrafik" title="Computergrafik">Computergrafik</a> angewendet.</li>

<li>Rotationsvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\alpha }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\alpha }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74b66e3c1118363d17c9e55e1858c8e402b25c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.343ex;" alt="{\displaystyle {\vec {\alpha }}}" loading="lazy"> sind in vielfältiger Weise definierbar, siehe die folgende Auflistung.</li></ul>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}{\vec {\alpha }}=\alpha {\hat {n}}&\rightarrow &R=I+{\frac {\sin(\alpha )}{\alpha }}[{\vec {\alpha }}]_{\times }+{\frac {1-\cos(\alpha )}{\alpha ^{2}}}[{\vec {\alpha }}]_{\times }^{2}=\exp([{\vec {\alpha }}]_{\times })\\{\vec {\alpha }}=\tan \left({\dfrac {\alpha }{2}}\right)\;{\hat {n}}&\rightarrow &R=I+{\dfrac {2}{1+{\vec {\alpha }}\cdot {\vec {\alpha }}}}([{\vec {\alpha }}]_{\times }+[{\vec {\alpha }}]_{\times }^{2})\\[2ex]{\vec {\alpha }}=\sin(\alpha )\;{\hat {n}}&\rightarrow &R=I+[{\vec {\alpha }}]_{\times }+{\dfrac {1}{1+\cos(\alpha )}}[{\vec {\alpha }}]_{\times }^{2}\\[2ex]{\vec {\alpha }}=\sin \left({\dfrac {\alpha }{2}}\right)\;{\hat {n}}&\rightarrow &R=I+2\cos \left({\dfrac {\alpha }{2}}\right)[{\vec {\alpha }}]_{\times }+2[{\vec {\alpha }}]_{\times }^{2}\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbd0272a98d3f0cdc46e331d862a3fae91c753d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:68.511ex; height:26.176ex;" alt="{\displaystyle {\begin{array}{lcl}{\vec {\alpha }}=\alpha {\hat {n}}&\rightarrow &R=I+{\frac {\sin(\alpha )}{\alpha }}[{\vec {\alpha }}]_{\times }+{\frac {1-\cos(\alpha )}{\alpha ^{2}}}[{\vec {\alpha }}]_{\times }^{2}=\exp([{\vec {\alpha }}]_{\times })\\{\vec {\alpha }}=\tan \left({\dfrac {\alpha }{2}}\right)\;{\hat {n}}&\rightarrow &R=I+{\dfrac {2}{1+{\vec {\alpha }}\cdot {\vec {\alpha }}}}([{\vec {\alpha }}]_{\times }+[{\vec {\alpha }}]_{\times }^{2})\\[2ex]{\vec {\alpha }}=\sin(\alpha )\;{\hat {n}}&\rightarrow &R=I+[{\vec {\alpha }}]_{\times }+{\dfrac {1}{1+\cos(\alpha )}}[{\vec {\alpha }}]_{\times }^{2}\\[2ex]{\vec {\alpha }}=\sin \left({\dfrac {\alpha }{2}}\right)\;{\hat {n}}&\rightarrow &R=I+2\cos \left({\dfrac {\alpha }{2}}\right)[{\vec {\alpha }}]_{\times }+2[{\vec {\alpha }}]_{\times }^{2}\end{array}}}" loading="lazy"></dd></dl>

Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy"> der <a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektor</a> in Richtung der <a href="Drehachse" class="mw-redirect" title="Drehachse">Drehachse</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\vec {\alpha }}]_{\times }}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [{\vec {\alpha }}]_{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8d65f347e8fca32ac6c43c2317c3d04537e6bf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.292ex; height:2.843ex;" alt="{\displaystyle [{\vec {\alpha }}]_{\times }}" loading="lazy"> ist die <a href="Kreuzprodukt#Kreuzproduktmatrix" title="Kreuzprodukt">Kreuzproduktmatrix</a> des Rotationsvektors. Die Auflistung gibt vier Darstellungen derselben Drehmatrix, die mit Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Drehmatrizen_des_Raumes_ℝⁿ">Drehmatrizen des Raumes ℝⁿ</h2></div>

Im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen Raum wird eine Drehung nicht durch eine Drehachse, sondern durch die Ebene definiert, die bei der Drehung auf sich selbst abgebildet wird. Das gilt auch in zwei Dimensionen, wo die Dreh-„Achse“ nur ein Punkt ist. Seien im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" 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}}" loading="lazy"> die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{1}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c309c6372acd1e6b4f1e33adb9f9167bd970e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.286ex; height:2.676ex;" alt="{\displaystyle {\hat {g}}_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{2}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44cd7906fe7f0e024ce2306b1b7dc8130c200f18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.286ex; height:2.676ex;" alt="{\displaystyle {\hat {g}}_{2}}" loading="lazy"> zwei zueinander orthogonale Einheitsvektoren (also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{1}\cdot {\hat {g}}_{2}=0}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{1}\cdot {\hat {g}}_{2}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b93dd346c0d1c765e4e0319ab1b90ee06dd84af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.512ex; height:2.676ex;" alt="{\displaystyle {\hat {g}}_{1}\cdot {\hat {g}}_{2}=0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\hat {g}}_{1}\right|=\left|{\hat {g}}_{2}\right|=1}">

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|{\hat {g}}_{1}\right|=\left|{\hat {g}}_{2}\right|=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c9801f6731d2ac4e7a11f4c844161b3180c2b99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.518ex; height:2.843ex;" alt="{\displaystyle \left|{\hat {g}}_{1}\right|=\left|{\hat {g}}_{2}\right|=1}" loading="lazy">), die demnach eine Ebene aufspannen. Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\hat {g}}_{1}\otimes {\hat {g}}_{1}+{\hat {g}}_{2}\otimes {\hat {g}}_{2}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\hat {g}}_{1}\otimes {\hat {g}}_{1}+{\hat {g}}_{2}\otimes {\hat {g}}_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a06a201a4e7dd5e882cac8c8df125b2d34e6d010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.55ex; height:2.676ex;" alt="{\displaystyle V={\hat {g}}_{1}\otimes {\hat {g}}_{1}+{\hat {g}}_{2}\otimes {\hat {g}}_{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\hat {g}}_{1}\otimes {\hat {g}}_{2}-{\hat {g}}_{2}\otimes {\hat {g}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle W={\hat {g}}_{1}\otimes {\hat {g}}_{2}-{\hat {g}}_{2}\otimes {\hat {g}}_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2d352a61b71846b491144d562a335d84941cb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.198ex; height:2.676ex;" alt="{\displaystyle W={\hat {g}}_{1}\otimes {\hat {g}}_{2}-{\hat {g}}_{2}\otimes {\hat {g}}_{1}}" loading="lazy">, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle I_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aba34f081d776e30204f3458e4f50b403b09e5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}" loading="lazy"> die Einheitsmatrix. Dann vermittelt die Matrix

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<annotation encoding="application/x-tex">{\displaystyle R=\exp(\alpha W)=I_{n}+\left(\cos(\alpha )-1\right)V+\sin(\alpha )W}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8028e9bf37eebcb1ac5275b4e359b9cc8ce24b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.15ex; height:2.843ex;" alt="{\displaystyle R=\exp(\alpha W)=I_{n}+\left(\cos(\alpha )-1\right)V+\sin(\alpha )W}" loading="lazy"></dd></dl>

eine Drehung um den Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{1}{\text{-}}{\hat {g}}_{2}{\text{-Ebene}}}">

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<mtext>-Ebene</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{1}{\text{-}}{\hat {g}}_{2}{\text{-Ebene}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae957ad8a23aaa0585b847b8260d6ab84aea4657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.354ex; height:2.676ex;" alt="{\displaystyle {\hat {g}}_{1}{\text{-}}{\hat {g}}_{2}{\text{-Ebene}}}" loading="lazy"> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(\alpha W\right):=\sum _{k=0}^{\infty }{\frac {\alpha ^{k}}{k\mathrm {!} }}{W}^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/698f764524db9535b3731b8bc12daa1389a64215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.781ex; height:7.009ex;" alt="{\displaystyle \exp \left(\alpha W\right):=\sum _{k=0}^{\infty }{\frac {\alpha ^{k}}{k\mathrm {!} }}{W}^{k}}" loading="lazy"></dd></dl>

und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{0}:=I_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle W^{0}:=I_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24a2fb8628e33b505260b114e26ceab3b91fa0be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.549ex; height:3.009ex;" alt="{\displaystyle W^{0}:=I_{n}}" loading="lazy"> definiert. Die Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(\alpha W)=I_{n}+\left(\cos(\alpha )-1\right)V+\sin(\alpha )W}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(\alpha W)=I_{n}+\left(\cos(\alpha )-1\right)V+\sin(\alpha )W}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a9ffd8e63967f1d4ca29922e668d0afc110ffee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.288ex; height:2.843ex;" alt="{\displaystyle \exp(\alpha W)=I_{n}+\left(\cos(\alpha )-1\right)V+\sin(\alpha )W}" loading="lazy"> ergibt sich aus den Identitäten

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{W}^{2}=&WW=-V\,,\quad WV=VW=W\,,\quad V^{2}=V\\\rightarrow W^{2n}=&(-1)^{n}V\quad {\text{und}}\quad {W}^{2n+1}=(-1)^{n}W\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1011bf464fe7b2a3ed9fcbcbdae6ef33a1b33b15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.707ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{W}^{2}=&WW=-V\,,\quad WV=VW=W\,,\quad V^{2}=V\\\rightarrow W^{2n}=&(-1)^{n}V\quad {\text{und}}\quad {W}^{2n+1}=(-1)^{n}W\end{aligned}}}" loading="lazy"></dd></dl>

sowie

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<annotation encoding="application/x-tex">{\displaystyle \cos(\alpha )=1+\sum _{k=1}^{\infty }{\frac {{(-1)}^{k}}{\left(2k\right)\mathrm {!} }}\alpha ^{2k}\quad {\text{und}}\quad \sin(\alpha )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)\mathrm {!} }}\alpha ^{2k+1}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c41a6c9491af3d5ecb85b1d1302fa778b746a16a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.072ex; height:7.509ex;" alt="{\displaystyle \cos(\alpha )=1+\sum _{k=1}^{\infty }{\frac {{(-1)}^{k}}{\left(2k\right)\mathrm {!} }}\alpha ^{2k}\quad {\text{und}}\quad \sin(\alpha )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)\mathrm {!} }}\alpha ^{2k+1}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Eigensystem_der_Drehmatrizen">Eigensystem der Drehmatrizen</h3></div>

Von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> wird jeder auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{1}}">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{1}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44cd7906fe7f0e024ce2306b1b7dc8130c200f18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.286ex; height:2.676ex;" alt="{\displaystyle {\hat {g}}_{2}}" loading="lazy"> senkrecht stehende Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

<mover>

</mover>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}\cdot {\hat {g}}_{1}={\vec {n}}\cdot {\hat {g}}_{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}\cdot {\hat {g}}_{1}={\vec {n}}\cdot {\hat {g}}_{2}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d86d94a235204a8348184492b0eb6202faab457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.079ex; height:2.843ex;" alt="{\displaystyle {\vec {n}}\cdot {\hat {g}}_{1}={\vec {n}}\cdot {\hat {g}}_{2}=0}" loading="lazy">) auf sich selbst abgebildet. Also sind diese Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"> Eigenvektoren von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> mit Eigenwert 1. Zwei Eigenwerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1,2}=e^{\pm \mathrm {i} \alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1,2}=e^{\pm \mathrm {i} \alpha }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60a2ae051e723cd3dfe5309e45c89697c3c5eb03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.891ex; height:3.343ex;" alt="{\displaystyle \lambda _{1,2}=e^{\pm \mathrm {i} \alpha }}" loading="lazy"> mit den Eigenvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {v}}_{1,2}={\tfrac {\sqrt {2}}{2}}\left({\hat {g}}_{1}\pm \mathrm {i} {\hat {g}}_{2}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {v}}_{1,2}={\tfrac {\sqrt {2}}{2}}\left({\hat {g}}_{1}\pm \mathrm {i} {\hat {g}}_{2}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2ee4b5fa6f3e014a52c367f085497eaca91d6a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.941ex; height:4.176ex;" alt="{\displaystyle {\hat {v}}_{1,2}={\tfrac {\sqrt {2}}{2}}\left({\hat {g}}_{1}\pm \mathrm {i} {\hat {g}}_{2}\right)}" loading="lazy">, worin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} ^{2}=-1}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{2}=-1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1903180d4ecaf05f4aef81a8ad389025f89c9d34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.77ex; height:2.843ex;" alt="{\displaystyle \mathrm {i} ^{2}=-1}" loading="lazy"> die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">imaginäre Einheit</a> ist. Aus diesen komplexen Eigenwerten und Eigenvektoren kann man also den Drehwinkel und die Drehebene rekonstruieren. Des Weiteren gilt bei Drehung in einer Ebene:

<mtr>

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<mi>arccos</mi>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Sp} R=&n+2\cos(\alpha )-2\rightarrow \alpha =\arccos \left({\frac {\operatorname {Sp} (R)+2-n}{2}}\right)\\R-{R}^{\mathsf {T}}=&2\sin(\alpha )W\rightarrow {\hat {g}}_{1}\otimes {\hat {g}}_{2}-{\hat {g}}_{2}\otimes {\hat {g}}_{1}=W={\frac {R-{R}^{\mathsf {T}}}{2\sin(\alpha )}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6634b95a5246e5794b7fd569129186e07623d77c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:60.192ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Sp} R=&n+2\cos(\alpha )-2\rightarrow \alpha =\arccos \left({\frac {\operatorname {Sp} (R)+2-n}{2}}\right)\\R-{R}^{\mathsf {T}}=&2\sin(\alpha )W\rightarrow {\hat {g}}_{1}\otimes {\hat {g}}_{2}-{\hat {g}}_{2}\otimes {\hat {g}}_{1}=W={\frac {R-{R}^{\mathsf {T}}}{2\sin(\alpha )}}\end{aligned}}}" loading="lazy"></dd></dl>

Allerdings kann eine Drehung im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen Raum gleichzeitig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2fd493036e5cdcb87f9ddfdc7ab475e74636b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.822ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{2}}}" loading="lazy"> (falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> gerade) oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n-1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n-1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3564ea5e9fdceafaa9d5070db79d7ce2278e7dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.923ex; height:3.676ex;" alt="{\displaystyle {\tfrac {n-1}{2}}}" loading="lazy"> (falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Allgemeine_Definition">Allgemeine Definition</h2></div>

Eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" 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}" loading="lazy">-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

<dl><dd>a) die Länge von Vektoren und die Winkel zwischen Vektoren erhält (ausgedrückt durch das <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a>), wenn also für alle Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle Rx,Ry\rangle =\langle x,y\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eeff2a5842df413234f62593b89c83609cce4ee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.283ex; height:2.843ex;" alt="{\displaystyle \langle Rx,Ry\rangle =\langle x,y\rangle }" loading="lazy"></dd></dl></dd></dl>

und

<dl><dd>b) <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientierungserhaltend</a> ist, wenn also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \,R=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \,R=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd2c3c758e42e7d7961d3dd00f6f5605d2f981ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.029ex; height:2.176ex;" alt="{\displaystyle \det \,R=1}" loading="lazy"> gilt.</dd></dl>

Drehmatrizen sind <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrizen</a> mit der <a href="Determinante" title="Determinante">Determinante</a> +1.

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

Weitere Eigenschaften von Rotationsmatrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\in \mathbb {R} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\in \mathbb {R} ^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/727eed959c2bf87e60cf6379b82a1c9401fba971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.766ex; height:2.343ex;" alt="{\displaystyle R\in \mathbb {R} ^{n\times n}}" loading="lazy">:

<ul><li><a href="Quadratische_Matrix" class="mw-redirect" title="Quadratische Matrix">Quadratische Matrix</a> mit <a href="Reelle_Zahl" title="Reelle Zahl">reellen</a> Komponenten</li></ul>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{\mathsf {T}}R=R\ R^{\mathsf {T}}={I_{n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4959be3a3780584fd6848d3dcd0857f7b05afa8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.778ex; height:3.009ex;" alt="{\displaystyle R^{\mathsf {T}}R=R\ R^{\mathsf {T}}={I_{n}}}" loading="lazy"> (<a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonal</a>), folgt aus dem ersten Teil der Definition:</li></ul>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\langle Rx,Ry\right\rangle \equiv \left\langle x,R^{\mathsf {T}}Ry\right\rangle =\left\langle x,y\right\rangle \quad \Rightarrow \quad R^{\mathsf {T}}R=I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bab765eaff193dcddef5f306c9822537ea42f8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.384ex; height:3.343ex;" alt="{\displaystyle \left\langle Rx,Ry\right\rangle \equiv \left\langle x,R^{\mathsf {T}}Ry\right\rangle =\left\langle x,y\right\rangle \quad \Rightarrow \quad R^{\mathsf {T}}R=I}" loading="lazy"></dd></dl></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{\mathsf {T}}=R^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85439c87cf2430c0c01e9eb25f6d6cee89ee3238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.311ex; height:2.676ex;" alt="{\displaystyle R^{\mathsf {T}}=R^{-1}}" loading="lazy"> (<a href="Transponierte_Matrix" title="Transponierte Matrix">Transponierte</a> und <a href="Inverse_Matrix" title="Inverse Matrix">Inverse</a> von R sind gleich), folgt aus der Orthogonalität.</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det(R)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ef4e4006d672c548a3bd9975f097cb817e5a87c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.064ex; height:2.843ex;" alt="{\displaystyle \det(R)=1}" loading="lazy"> (<a href="Determinante_(Mathematik)" class="mw-redirect" title="Determinante (Mathematik)">Determinante</a>), entspricht dem zweiten Teil der Definition.</li></ul>

<ul><li>Die Ausrichtung des Koordinatensystems (<a href="Rechtssystem_(Mathematik)" title="Rechtssystem (Mathematik)">Rechts-</a> oder Linkssystem) wird beibehalten, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(R)=1>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det(R)=1>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13214a74024ff311f9ecb764f97765ccc6f67a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.325ex; height:2.843ex;" alt="{\displaystyle \det(R)=1>0}" loading="lazy"> <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">positive Orientierung</a>.</li></ul>

<ul><li>Die Kombination einer Drehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1d63c96f59d98589d923c4f0b04222feaa7283e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{1}}" loading="lazy"> mit anschließender Drehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35f571121c264178676d1df8ab899f238a39bc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{2}}" loading="lazy"> erfolgt mit der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2}R_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{2}R_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcd9c2437792e882c7f66a74021cb374e569224f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.636ex; height:2.509ex;" alt="{\displaystyle R_{2}R_{1}}" loading="lazy">. Weil die <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikation</a> nicht <a href="Kommutativgesetz" title="Kommutativgesetz">kommutativ</a> ist, führt die umgekehrte Reihenfolge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}R_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{1}R_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a54b43c632e9d799fd1e435335763a26db9453c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.636ex; height:2.509ex;" alt="{\displaystyle R_{1}R_{2}}" loading="lazy"> im Allgemeinen zu einem anderen Ergebnis. Nur bei infinitesimal kleinen Drehungen ist die Reihenfolge vertauschbar, siehe <a href="#Kommutativität_infinitesimaler_Drehungen">#Kommutativität infinitesimaler Drehungen</a>.</li></ul>

<ul><li>Die Menge aller Drehmatrizen eines Raumes bildet die <a href="Drehgruppe" title="Drehgruppe">Drehgruppe</a>, nämlich die <a href="Spezielle_orthogonale_Gruppe" class="mw-redirect" title="Spezielle orthogonale Gruppe">spezielle orthogonale Gruppe</a>:</li></ul>

</mrow>

<mrow>

<mrow>

<mtext>lineare Abbildung </mtext>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)=\left\{{\text{lineare Abbildung }}R\colon \,\mathbb {R} ^{n}\to \mathbb {R} ^{n}\ |\ R^{\mathsf {T}}R=I_{n}\,,\ \det \,R=1\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11b74c77beff70cc2fdb7de95cd397729a04d427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.699ex; height:3.343ex;" alt="{\displaystyle \mathrm {SO} (n)=\left\{{\text{lineare Abbildung }}R\colon \,\mathbb {R} ^{n}\to \mathbb {R} ^{n}\ |\ R^{\mathsf {T}}R=I_{n}\,,\ \det \,R=1\right\}}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Zusätzlich zur <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraischen Struktur</a> einer <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> besitzt die Menge aller Drehmatrizen auch eine <a href="Topologische_Struktur" class="mw-redirect" title="Topologische Struktur">topologische Struktur</a>: Die Operationen Multiplikation und Inversion von Drehmatrizen sind stetig differenzierbare Funktionen ihrer Parameter, der Drehwinkel. Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa71842f19b6810b4bfa9eb282e92fbf285094e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n)}" loading="lazy"> bildet eine <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbare Mannigfaltigkeit</a> und ist somit eine <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a>. Diese hat die <a href="Dimension_(Mathematik)#Dimension_einer_Mannigfaltigkeit" title="Dimension (Mathematik)">Dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(n-1)/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n(n-1)/2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b1d96c185de1bffc1e78739934b09489f683efc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.926ex; height:2.843ex;" alt="{\displaystyle n(n-1)/2}" loading="lazy">.</li></ul>

<ul><li>Mit der Lie-Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57c18bd074c7bdc79ea650563edde2e2bc080321.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.412ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(n)}" loading="lazy"> verknüpft, ein Vektorraum mit einem <a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilinearen</a> alternierenden Produkt (<a href="Lie-Klammer" title="Lie-Klammer">Lie-Klammer</a>), wobei der Vektorraum bezüglich der Lie-Klammer abgeschlossen ist. Dieser Vektorraum ist <a href="Isomorphismus" title="Isomorphismus">isomorph</a> zum <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> am neutralen Element der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa71842f19b6810b4bfa9eb282e92fbf285094e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n)}" loading="lazy"> (neutrales Element ist die Einheitsmatrix), sodass insbesondere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim {\mathfrak {so}}(n)=\dim \mathrm {SO} (n)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim {\mathfrak {so}}(n)=\dim \mathrm {SO} (n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/276c073099c88e887edba2e3c6f4c8d8e87bce53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.299ex; height:2.843ex;" alt="{\displaystyle \dim {\mathfrak {so}}(n)=\dim \mathrm {SO} (n)}" loading="lazy"> gilt. Die Lie-Algebra besteht aus allen <a href="Schiefsymmetrische_Matrix" title="Schiefsymmetrische Matrix">schiefsymmetrischen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" 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}" loading="lazy">-Matrizen und ihre Basis sind die sog. Erzeugenden. Die <a href="Exponentialabbildung" title="Exponentialabbildung">Exponentialabbildung</a> verknüpft die Lie-Algebra mit der Lie-Gruppe:</li></ul>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \colon \ {\mathfrak {so}}(n)\to \mathrm {SO} (n),\ J\mapsto \sum _{k=0}^{\infty }{\frac {1}{k!}}J^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5872b4b711cf8f00d36f794531401087db6d48a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.208ex; height:7.009ex;" alt="{\displaystyle \exp \colon \ {\mathfrak {so}}(n)\to \mathrm {SO} (n),\ J\mapsto \sum _{k=0}^{\infty }{\frac {1}{k!}}J^{k}}" loading="lazy"></dd></dl></dd></dl>

Speziell bei Drehungen in einer Ebene gilt für Rotationsmatrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\in \mathbb {R} ^{n\times n}}">

<msup>

</mrow>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R\in \mathbb {R} ^{n\times n}}</annotation>

</semantics>

<msup>

</mrow>

</msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R^{-1}(\alpha )=R({-\alpha })=R(2\pi -\alpha )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc97ac0f92bc26c5cb02757c71986c26b4564ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.855ex; height:3.176ex;" alt="{\displaystyle R^{-1}(\alpha )=R({-\alpha })=R(2\pi -\alpha )}" loading="lazy"></li></ul>

<ul><li>Zwei Vektoren spannen die Drehebene auf und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff40d66ad535411eedb9c686a9008a5089c35ac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-2}" loading="lazy"> Vektoren werden von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

<ul><li>Die zur Drehebene senkrechten Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation>

</semantics>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (R-I){\vec {v}}={\vec {0}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/650a34d793bd0473fef23801ea972dc72acc20fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.669ex; height:3.343ex;" alt="{\displaystyle (R-I){\vec {v}}={\vec {0}}.}" loading="lazy"></dd></dl></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (R-I)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20d1d1d0c33252b04992d48c473c4e711d77ffd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.585ex; height:2.843ex;" alt="{\displaystyle (R-I)}" loading="lazy"> für ungerade Dimensionen nicht <a href="Regul%C3%A4re_Matrix" title="Reguläre Matrix">regulär</a> ist, ist die Berechnung dieser Vektoren über eine <a href="Eigenwertzerlegung" class="mw-redirect" title="Eigenwertzerlegung">Eigenwertzerlegung</a> durchzuführen. Die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"> sind <a href="Eigenvektor" class="mw-redirect" title="Eigenvektor">Eigenvektor</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> mit <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> 1. In geraden Dimensionen muss kein Eigenvektor zum Eigenwert 1 existieren, was im Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy"> anschaulich klar ist.</dd></dl>

<ul><li>Der Drehwinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \left\langle {\vec {w}},R{\vec {w}}\right\rangle =\left\|{\vec {w}}\right\|\left\|R{\vec {w}}\right\|\cos \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58906423a14beb18312a9451a68746e6ffd7bd6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.859ex; height:2.843ex;" alt="{\displaystyle \quad \left\langle {\vec {w}},R{\vec {w}}\right\rangle =\left\|{\vec {w}}\right\|\left\|R{\vec {w}}\right\|\cos \alpha }" loading="lazy"></dd></dl></dd>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {w}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}}" loading="lazy"> in der Drehebene, in drei Dimensionen also orthogonal zur Drehachse, oder aus der <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> der Drehmatrix

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spur} (R)=n-2+2\cos \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75608d8a5835cc8c311ddb66b150c6fb59f2fcbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.234ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spur} (R)=n-2+2\cos \alpha }" loading="lazy"></dd></dl></dd>

<dd>(siehe auch Formel für die Matrix einer Drehung um eine allgemeine Achse oben).</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Infinitesimale_Drehungen">Infinitesimale Drehungen</h2></div>

Betrachtet man Drehungen um infinitesimal kleine Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \alpha }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43f93ed80dbac5853af16c1a6ebd77997a443491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \alpha }" loading="lazy">, so ist es ausreichend, die Winkelfunktionen der endlichen Drehung bis zur ersten Ordnung zu entwickeln (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17e6c4959a2bf1e6f704c4f0a29a34b8edf25c12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.001ex; height:2.176ex;" alt="{\displaystyle \sin x=x}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/691abe771f0a6d98899f0562259a0d71207b1866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.089ex; height:2.176ex;" alt="{\displaystyle \cos x=1}" loading="lazy">). Damit lassen sich nun infinitesimale Drehungen darstellen als

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(\mathrm {d} \alpha )=I+\mathrm {d} \alpha \,J,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/698fcb71a8cf0d7f529df6a4e01283a9f9f2b71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.749ex; height:2.843ex;" alt="{\displaystyle R(\mathrm {d} \alpha )=I+\mathrm {d} \alpha \,J,}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> die Einheitsmatrix und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> die Erzeugende einer infinitesimalen Drehung darstellt. Die Erzeugenden sind die Ableitungen der Rotationsmatrix an der Stelle der Identität und bilden die Basis der Lie-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n)}</annotation>

</semantics>

<msub>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J=\left.{\frac {\mathrm {d} R(\alpha )}{\mathrm {d} \alpha }}\right|_{\alpha =0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbc617b6cd0b0d57d1047440a85d4263cd1c1003.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.791ex; height:6.509ex;" alt="{\displaystyle J=\left.{\frac {\mathrm {d} R(\alpha )}{\mathrm {d} \alpha }}\right|_{\alpha =0}}" loading="lazy"></dd></dl>

Eine endliche Drehung lässt sich durch Hintereinanderausführung infinitesimaler Drehungen erzeugen:

</mrow>

</munder>

<msup>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</munder>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(\alpha )=\lim _{N\to \infty }\left[R\left({\frac {\alpha }{N}}\right)\right]^{N}=\lim _{N\to \infty }\left[I+{\frac {\alpha }{N}}\,J\right]^{N}=\exp \left(\alpha J\right)\equiv \sum _{n=0}^{\infty }{\frac {\left(\alpha J\right)^{n}}{n!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f4b6bd26159ce5c3474d6e5e120ddfa71bc1652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.975ex; height:6.843ex;" alt="{\displaystyle R(\alpha )=\lim _{N\to \infty }\left[R\left({\frac {\alpha }{N}}\right)\right]^{N}=\lim _{N\to \infty }\left[I+{\frac {\alpha }{N}}\,J\right]^{N}=\exp \left(\alpha J\right)\equiv \sum _{n=0}^{\infty }{\frac {\left(\alpha J\right)^{n}}{n!}}}" loading="lazy"></dd></dl>

Dabei wurde die <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> identifiziert. Die Exponentialfunktion von Matrizen ist über die Reihendarstellung definiert, wie im letzten Schritt gezeigt. Es lässt sich zeigen, dass Erzeugende <a href="Spur_(Mathematik)" title="Spur (Mathematik)">spurfrei</a> sein müssen:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=\det R(\alpha )=\exp(\alpha \ \operatorname {Sp} \,J)\quad \implies \quad \operatorname {Sp} \,J=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03514a83e93715065ee9a4b17f185be7cf0c5002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.808ex; height:2.843ex;" alt="{\displaystyle 1=\det R(\alpha )=\exp(\alpha \ \operatorname {Sp} \,J)\quad \implies \quad \operatorname {Sp} \,J=0}" loading="lazy"></dd></dl>

und schiefsymmetrisch sind:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=R(\alpha )R^{\mathsf {T}}(\alpha )=R^{\mathsf {T}}(\alpha )R(\alpha )=e^{\alpha J}e^{\alpha J^{\mathsf {T}}}=e^{\alpha J^{\mathsf {T}}}e^{\alpha J}=e^{\alpha (J+J^{\mathsf {T}})}\quad \implies \quad J+J^{\mathsf {T}}=0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c49a934eb8830834940c91312fe8d38c05b76d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:86.712ex; height:3.509ex;" alt="{\displaystyle I=R(\alpha )R^{\mathsf {T}}(\alpha )=R^{\mathsf {T}}(\alpha )R(\alpha )=e^{\alpha J}e^{\alpha J^{\mathsf {T}}}=e^{\alpha J^{\mathsf {T}}}e^{\alpha J}=e^{\alpha (J+J^{\mathsf {T}})}\quad \implies \quad J+J^{\mathsf {T}}=0.}" loading="lazy"></dd></dl>

Mit dem Konzept der Erzeugenden lässt sich die lokale Gruppenstruktur der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa71842f19b6810b4bfa9eb282e92fbf285094e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n)}" loading="lazy"> in der Umgebung der <a href="Identische_Abbildung" title="Identische Abbildung">identischen Abbildung</a> ausdrücken, und zwar durch die infinitesimalen Drehungen. Wegen des Zusammenhangs über die Exponentialfunktion wird aus einer Multiplikation von Drehmatrizen eine Addition ihrer Erzeugenden. Die Erzeugenden bilden einen Vektorraum derselben Dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=n(n-1)/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=n(n-1)/2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78aed80ed133b5975855ef7a6e48b3133319115b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.852ex; height:2.843ex;" alt="{\displaystyle G=n(n-1)/2}" loading="lazy"> wie die Drehgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation>

</semantics>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> linear unabhängige Erzeugende der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation>

</semantics>

Die Erzeugenden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07a98dbd556141d7f0a16eed84fe0ae6467dcd23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.09ex; height:2.509ex;" alt="{\displaystyle J_{i}}" loading="lazy"> bilden mit dem Lie-Produkt (<a href="Kommutator_(Mathematik)" title="Kommutator (Mathematik)">Kommutator</a>) die sog. <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n)}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [J_{i},J_{k}]=\sum _{l}c_{ik}^{l}J_{l}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aebb35710af0471a9bd656be04a9b3378b1af84e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.313ex; height:5.509ex;" alt="{\displaystyle [J_{i},J_{k}]=\sum _{l}c_{ik}^{l}J_{l}}" loading="lazy"></dd></dl>

Die Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{ik}^{l}=-c_{ki}^{l}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{ik}^{l}=-c_{ki}^{l}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/807286c3afa822ac1a74133c69d99181e486d92b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.233ex; height:3.176ex;" alt="{\displaystyle c_{ik}^{l}=-c_{ki}^{l}}" loading="lazy"> sind charakteristische Konstanten der Gruppe. Für alle doppelten Kommutatoren gilt die <a href="Jacobi-Identit%C3%A4t" title="Jacobi-Identität">Jacobi-Identität</a>:

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[[J_{i},J_{k}],J_{l}\right]+\left[[J_{k},J_{l}],J_{i}\right]+\left[[J_{l},J_{i}],J_{k}\right]=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54c0c990feee6af672ddbdbc9499facf48aaabb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.352ex; height:2.843ex;" alt="{\displaystyle \left[[J_{i},J_{k}],J_{l}\right]+\left[[J_{k},J_{l}],J_{i}\right]+\left[[J_{l},J_{i}],J_{k}\right]=0}" loading="lazy"></dd></dl>

In der theoretischen Physik spielen Lie-Gruppen eine wichtige Rolle, z. B. in der Quantenmechanik (siehe <a href="Drehimpulsoperator" class="mw-redirect" title="Drehimpulsoperator">Drehimpulsoperator</a>) oder der <a href="Elementarteilchenphysik" class="mw-redirect" title="Elementarteilchenphysik">Elementarteilchenphysik</a>.

<div class="mw-heading mw-heading3"><h3 id="Ebene_ℝ²">Ebene ℝ²</h3></div>

Für Drehungen im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

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</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle R(\mathrm {d} \alpha )={\begin{pmatrix}1&-\mathrm {d} \alpha \\\mathrm {d} \alpha &1\end{pmatrix}}\,,\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7166c4028f098f8b63d1aa79bc006717dd65e05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.257ex; height:6.176ex;" alt="{\displaystyle R(\mathrm {d} \alpha )={\begin{pmatrix}1&-\mathrm {d} \alpha \\\mathrm {d} \alpha &1\end{pmatrix}}\,,\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}" loading="lazy"></dd></dl>

Für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (2)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f580f01b0e7df995ae24684960f9f4b3487ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (2)}" loading="lazy"> gibt es nur eine linear unabhängige Erzeugende.

Eine endliche Drehung lässt sich über die Exponentialfunktion des Drehwinkels und der Erzeugenden darstellen. Dies wird hier auf eine weitere Art gezeigt: Die Drehmatrix wird in einen <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrischen</a> und antisymmetrischen Anteil zerlegt und die trigonometrischen Funktionen werden durch ihre Taylorreihe dargestellt.

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<annotation encoding="application/x-tex">{\displaystyle R(\alpha )=I\,\cos \alpha +J\,\sin \alpha =I\,\sum _{n=0}^{\infty }(-1)^{n}{\frac {\alpha ^{2n}}{(2n)!}}+J\,\sum _{n=0}^{\infty }(-1)^{n}{\frac {\alpha ^{2n+1}}{(2n+1)!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59fe3fccead50cd4990f4731fd59820519c42c8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.446ex; height:6.843ex;" alt="{\displaystyle R(\alpha )=I\,\cos \alpha +J\,\sin \alpha =I\,\sum _{n=0}^{\infty }(-1)^{n}{\frac {\alpha ^{2n}}{(2n)!}}+J\,\sum _{n=0}^{\infty }(-1)^{n}{\frac {\alpha ^{2n+1}}{(2n+1)!}}}" loading="lazy"></dd></dl>

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}=-I}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle J^{2}=-I}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b270535a7c53021b4f1e45e59c2fcdd7807f33cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.658ex; height:2.843ex;" alt="{\displaystyle J^{2}=-I}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2n}=(-I)^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle J^{2n}=(-I)^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4036f5d152ddf75c5341235d5038adff76d6b678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.672ex; height:3.176ex;" alt="{\displaystyle J^{2n}=(-I)^{n}}" loading="lazy"> folgt das von oben bekannte Ergebnis:

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<annotation encoding="application/x-tex">{\displaystyle R(\alpha )=\sum _{n=0}^{\infty }J^{2n}{\frac {\alpha ^{2n}}{(2n)!}}+\sum _{n=0}^{\infty }J^{2n+1}{\frac {\alpha ^{2n+1}}{(2n+1)!}}=\exp(\alpha J)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a143c563b95a60ed4ff2c3734ab7f650b002b901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.838ex; height:6.843ex;" alt="{\displaystyle R(\alpha )=\sum _{n=0}^{\infty }J^{2n}{\frac {\alpha ^{2n}}{(2n)!}}+\sum _{n=0}^{\infty }J^{2n+1}{\frac {\alpha ^{2n+1}}{(2n+1)!}}=\exp(\alpha J)}" loading="lazy"></dd></dl>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{x}(\mathrm {d} \alpha )&={\begin{pmatrix}1&0&0\\0&1&-\mathrm {d} \alpha \\0&\mathrm {d} \alpha &1\end{pmatrix}}\,,\quad &J_{x}&={\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix}}\\R_{y}(\mathrm {d} \alpha )&={\begin{pmatrix}1&0&\mathrm {d} \alpha \\0&1&0\\-\mathrm {d} \alpha &0&1\end{pmatrix}}\,,\quad &J_{y}&={\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix}}\\R_{z}(\mathrm {d} \alpha )&={\begin{pmatrix}1&-\mathrm {d} \alpha &0\\\mathrm {d} \alpha &1&0\\0&0&1\end{pmatrix}}\,,\quad &J_{z}&={\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa220aee7c3b6d36320a9d7850984e69c96f3540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:57.692ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}R_{x}(\mathrm {d} \alpha )&={\begin{pmatrix}1&0&0\\0&1&-\mathrm {d} \alpha \\0&\mathrm {d} \alpha &1\end{pmatrix}}\,,\quad &J_{x}&={\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix}}\\R_{y}(\mathrm {d} \alpha )&={\begin{pmatrix}1&0&\mathrm {d} \alpha \\0&1&0\\-\mathrm {d} \alpha &0&1\end{pmatrix}}\,,\quad &J_{y}&={\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix}}\\R_{z}(\mathrm {d} \alpha )&={\begin{pmatrix}1&-\mathrm {d} \alpha &0\\\mathrm {d} \alpha &1&0\\0&0&1\end{pmatrix}}\,,\quad &J_{z}&={\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}}\end{aligned}}}" loading="lazy"></dd></dl>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (3)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8366fc6e92660ba077b87b745b305a4176b1d1ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (3)}" loading="lazy"> gibt es drei linear unabhängige Erzeugende. Anders als endliche Drehungen vertauschen infinitesimale Drehungen miteinander (der <a href="Kommutator_(Mathematik)" title="Kommutator (Mathematik)">Kommutator</a> verschwindet in erster Ordnung in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \alpha }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \alpha }</annotation>

</semantics>

Eine infinitesimale Drehung und ihre Erzeugende um eine beliebige Achse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}\cdot {\hat {n}}=1}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle R_{\hat {n}}(\mathrm {d} \alpha )=I+\mathrm {d} \alpha \sum _{i}({\hat {n}}\times {\hat {e}}_{i})\otimes {\hat {e}}_{i}={\begin{pmatrix}1&-\mathrm {d} \alpha \,n_{z}&\mathrm {d} \alpha \,n_{y}\\\mathrm {d} \alpha \,n_{z}&1&-\mathrm {d} \alpha \,n_{x}\\-\mathrm {d} \alpha \,n_{y}&\mathrm {d} \alpha \,n_{x}&1\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/283fced09d49d92c4042d4a9bd3d83a433bde054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:69.168ex; height:9.843ex;" alt="{\displaystyle R_{\hat {n}}(\mathrm {d} \alpha )=I+\mathrm {d} \alpha \sum _{i}({\hat {n}}\times {\hat {e}}_{i})\otimes {\hat {e}}_{i}={\begin{pmatrix}1&-\mathrm {d} \alpha \,n_{z}&\mathrm {d} \alpha \,n_{y}\\\mathrm {d} \alpha \,n_{z}&1&-\mathrm {d} \alpha \,n_{x}\\-\mathrm {d} \alpha \,n_{y}&\mathrm {d} \alpha \,n_{x}&1\end{pmatrix}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle J_{\hat {n}}=\sum _{i}\left({\hat {n}}\times {\hat {e}}_{i}\right)\otimes {\hat {e}}_{i}={\begin{pmatrix}0&-n_{z}&n_{y}\\n_{z}&0&-n_{x}\\-n_{y}&n_{x}&0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1b1e8e98acc514a7d1ff6b3fcd8c7501671c632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:47.811ex; height:9.843ex;" alt="{\displaystyle J_{\hat {n}}=\sum _{i}\left({\hat {n}}\times {\hat {e}}_{i}\right)\otimes {\hat {e}}_{i}={\begin{pmatrix}0&-n_{z}&n_{y}\\n_{z}&0&-n_{x}\\-n_{y}&n_{x}&0\end{pmatrix}}}" loading="lazy"></dd></dl>

Hieran sieht man, dass eine beliebige Erzeugende stets eine schiefsymmetrische Matrix ist.

Eine endliche Drehung um eine beliebige Achse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}\cdot {\hat {n}}=1}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle R_{\hat {n}}(\alpha )=\exp {\Big (}\alpha \,J_{\hat {n}}{\Big )}=\exp {\Big (}\alpha \,{\hat {n}}\cdot {\vec {J}}\,{\Big )}=\exp {\Big (}\alpha (n_{x}J_{x}+n_{y}J_{y}+n_{z}J_{z}){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e431844c27c06c70cd46c502117ff86b5adb22ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:70.707ex; height:4.843ex;" alt="{\displaystyle R_{\hat {n}}(\alpha )=\exp {\Big (}\alpha \,J_{\hat {n}}{\Big )}=\exp {\Big (}\alpha \,{\hat {n}}\cdot {\vec {J}}\,{\Big )}=\exp {\Big (}\alpha (n_{x}J_{x}+n_{y}J_{y}+n_{z}J_{z}){\Big )}}" loading="lazy"></dd></dl>

Die Erzeugenden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/573c4eac7701ac45330a1677abe1c3a7fc054ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle J_{x}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{y}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{y}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/782c237016b7230689619e010bf692a0adc79b38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.34ex; height:2.843ex;" alt="{\displaystyle J_{y}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e4fc97aafeb0dc8febc33638f7644e68c2f68ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle J_{z}}" loading="lazy"> bilden die sog. Lie-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(3)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}" loading="lazy">, d. h., der Kommutator (Lie-Produkt) zweier Erzeugenden liegt wieder in der Menge der Erzeugenden:

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<annotation encoding="application/x-tex">{\displaystyle [J_{x},J_{y}]=J_{z}\,,\quad [J_{x},J_{z}]=-J_{y}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbbcf21b49474b8d543f27d1a5d341e685e735df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.592ex; height:3.009ex;" alt="{\displaystyle [J_{x},J_{y}]=J_{z}\,,\quad [J_{x},J_{z}]=-J_{y}}" loading="lazy"></dd></dl>

und ebenso für alle <a href="Zyklische_Permutation" title="Zyklische Permutation">zyklischen Permutationen</a> der Indizes.

<div class="mw-heading mw-heading3"><h3 id="Kommutativität_infinitesimaler_Drehungen">Kommutativität infinitesimaler Drehungen</h3></div>

<div class="sieheauch" role="navigation" style="font-style:italic;">Siehe auch: <a href="Winkelgeschwindigkeit#Kommutativität" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit#Kommutativität</a></div>

Zwei infinitesimale Drehungen sind in ihrer Reihenfolge vertauschbar, was bei großen Drehungen im Allgemeinen nicht der Fall ist, siehe <a href="#Eigenschaften">#Eigenschaften</a>. Ersichtlich ist das am Produkt zweier infinitesimaler Drehungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}=I+\mathrm {d} \alpha J_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{n}=I+\mathrm {d} \alpha J_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c45d913258cf85a82365385fb575de590c944060.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.382ex; height:2.509ex;" alt="{\displaystyle R_{n}=I+\mathrm {d} \alpha J_{n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{m}=I+\mathrm {d} \beta J_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{m}=I+\mathrm {d} \beta J_{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c1d975786cbc0c9d6d90c45f6ea19eee3d0b927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.139ex; height:2.509ex;" alt="{\displaystyle R_{m}=I+\mathrm {d} \beta J_{m}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{n}R_{m}=&(I+\mathrm {d} \alpha J_{n})(I+\mathrm {d} \beta J_{m})\\=&I+\mathrm {d} \alpha J_{n}+\mathrm {d} \beta J_{m}+\mathrm {d} \alpha \mathrm {d} \beta J_{n}J_{m}\\\approx &I+\mathrm {d} \alpha J_{n}+\mathrm {d} \beta J_{m}\approx I+\mathrm {d} \alpha J_{n}+\mathrm {d} \beta J_{m}+\mathrm {d} \alpha \mathrm {d} \beta J_{m}J_{n}\\=&(I+\mathrm {d} \beta J_{m})(I+\mathrm {d} \alpha J_{n})=R_{m}R_{n},\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee57247a64ffa4dcfcb5ed432c0ea52cb5d3060e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:61.906ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}R_{n}R_{m}=&(I+\mathrm {d} \alpha J_{n})(I+\mathrm {d} \beta J_{m})\\=&I+\mathrm {d} \alpha J_{n}+\mathrm {d} \beta J_{m}+\mathrm {d} \alpha \mathrm {d} \beta J_{n}J_{m}\\\approx &I+\mathrm {d} \alpha J_{n}+\mathrm {d} \beta J_{m}\approx I+\mathrm {d} \alpha J_{n}+\mathrm {d} \beta J_{m}+\mathrm {d} \alpha \mathrm {d} \beta J_{m}J_{n}\\=&(I+\mathrm {d} \beta J_{m})(I+\mathrm {d} \alpha J_{n})=R_{m}R_{n},\end{aligned}}}" loading="lazy"></dd></dl>

denn die Terme, die proportional zum Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \alpha \mathrm {d} \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \alpha \mathrm {d} \beta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40fae8a24ca1a58fb696438256a11f5c4ec57375.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \alpha \mathrm {d} \beta }" loading="lazy"> zweier infinitesimaler Größen sind, können gegenüber den anderen vernachlässigt werden.

<div class="mw-heading mw-heading2"><h2 id="Bestimmung_der_Drehung_zwischen_zwei_Lagen">Bestimmung der Drehung zwischen zwei Lagen</h2></div>

Gegeben sei die Lage eines Körpers in zwei Positionen. Außerdem sei die Positionsänderung durch Drehung um den Ursprung erfolgt. Gesucht ist die oder eine Drehmatrix, die diese Drehung beschreibt. Im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen Raum wird die Lage des Körpers durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}_{i},\;i=1\ldots n}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}_{i},\;i=1\ldots n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ef03ecf63649c4091606cbd2547ef661b11aee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.764ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}_{i},\;i=1\ldots n}" loading="lazy"> beschrieben, welche die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\Big (}{\vec {x}}_{1}\ldots {\vec {x}}_{n}{\Big )}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95b35631166033b05ba38e9a1e299fe99d6db044.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.283ex; height:4.843ex;" alt="{\displaystyle X={\Big (}{\vec {x}}_{1}\ldots {\vec {x}}_{n}{\Big )}}" loading="lazy"> bilden. Die Ausgangslage werde durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle R\,X_{0}=X.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15dbe4727e645a49dc016d920a034937d80c6133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.855ex; height:2.509ex;" alt="{\displaystyle R\,X_{0}=X.}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6eef7135bb088ebb34800e29de5060d7dc0ed804.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.33ex; height:3.343ex;" alt="{\displaystyle X_{0}^{-1}}" loading="lazy"> bestimmt werden:

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<annotation encoding="application/x-tex">{\displaystyle R=X\,X_{0}^{-1}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0df43e8e2f03a67b0c937a1ae4fd766644abb98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.206ex; height:3.343ex;" alt="{\displaystyle R=X\,X_{0}^{-1}.}" loading="lazy"></dd></dl>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6381fdad2b9f11954b1fc2db08bbaccf634ededa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}" loading="lazy"> nicht regulär, weil zum Beispiel einer der Punkte des Körpers im Ursprung liegt, dann kann die Inverse nicht gebildet werden. Auch die <a href="Pseudoinverse" title="Pseudoinverse">Pseudoinverse</a> führt hier nicht zum Ziel. Allerdings kann eine <a href="Singul%C3%A4rwertzerlegung" title="Singulärwertzerlegung">Singulärwertzerlegung</a> durchgeführt werden. Diese liefert für eine Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> die <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitären</a> Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"> sowie die Diagonalmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e1f558f53cda207614abdf90162266c70bc5c1e.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 \Sigma }" loading="lazy"> der Singulärwerte:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(U,\Sigma ,V)&={\text{svd}}(X)\\X&=U\,\Sigma \,V^{*}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3a477b4d082a97711c3ec83bb4729946a8dce8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.201ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}(U,\Sigma ,V)&={\text{svd}}(X)\\X&=U\,\Sigma \,V^{*}\end{aligned}}}" loading="lazy"></dd></dl>

Man kann zeigen, dass die Singulärwerte gegenüber einer Rotation invariant sind. Es gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma =\Sigma _{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd69c88ff30f10c6cbca8c38a83247c1cecf2b13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.509ex; height:2.509ex;" alt="{\displaystyle \Sigma =\Sigma _{0}}" loading="lazy"> und damit

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R\,X_{0}&=X\\R\,U_{0}\,\Sigma _{0}\,V_{0}^{*}&=U\,\Sigma \,V^{*}\\R&=U\,V^{*}\,V_{0}\,U_{0}^{*}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48499573faceef941986b6c547f1a682e0d7bd5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.887ex; margin-bottom: -0.285ex; width:26.987ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}R\,X_{0}&=X\\R\,U_{0}\,\Sigma _{0}\,V_{0}^{*}&=U\,\Sigma \,V^{*}\\R&=U\,V^{*}\,V_{0}\,U_{0}^{*}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Orthogonaler_Tensor" title="Orthogonaler Tensor">Orthogonaler Tensor</a></li>

<li><a href="Orthogonale_Abbildung" title="Orthogonale Abbildung">Orthogonale Abbildung</a></li></ul>

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

<ul><li><a href="Gerd_Fischer_(Mathematiker)" title="Gerd Fischer (Mathematiker)">Gerd Fischer</a>: Lineare Algebra. Eine Einführung für Studienanfänger. 17. aktualisierte Auflage. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 (Studium. Grundkurs Mathematik).</li>

<li>Karlheinz Goldhorn: Moderne mathematische Methoden der Physik. Band 2. Springer, Berlin u. a. 2010, ISBN 978-3-642-05184-5 (Springer-Lehrbuch).</li>

<li><a href="Max_Koecher" title="Max Koecher">Max Koecher</a>: Lineare Algebra und analytische Geometrie. 4. ergänzte und aktualisierte Auflage. Springer, Berlin u. a. 1997, ISBN 3-540-62903-3 (Grundwissen Mathematik – Springer-Lehrbuch).</li>

<li><a href="Florian_Scheck" title="Florian Scheck">Florian Scheck</a>: Theoretische Physik. Band 1: Mechanik, von den Newtonschen Gesetzen zum deterministischen Chaos. 8. Auflage. Springer, Berlin u. a. 2007, ISBN 978-3-540-71377-7.</li>

<li>J. Hanson: Rotations in three, four, and five dimensions. <a href="ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1103.5263">1103.5263</a>.</li></ul>

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

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RotationMatrix.html">Rotation Matrix.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-23" href="https://de.wikipedia.org/wiki/?title=Drehmatrix&oldid=262666487">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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