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<title>Involution (Mathematik)</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Involution (Mathematik)</span></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"><p><b>Involution</b> bedeutet in der Mathematik eine selbst<a href="Inverses_Element" title="Inverses Element">inverse</a> <a href="Abbildung_(Mathematik)" class="mw-redirect" title="Abbildung (Mathematik)">Abbildung</a>. Die Bezeichnung leitet sich von dem lateinischen Wort <i>involvere</i> „einwickeln“ ab.

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

<p>Eine Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\rightarrow A}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon A\rightarrow A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bddfc0627eb12b864492f45cce430616f7a2a95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.413ex; height:2.509ex;" alt="{\displaystyle f\colon A\rightarrow A}" loading="lazy"></span> mit übereinstimmender <a href="Definitionsmenge" title="Definitionsmenge">Definitions-</a> und <a href="Zielmenge" title="Zielmenge">Zielmenge</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> heißt <i>Involution</i>, wenn für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27bcc9b2afb295d4234bc294860cd0c63bcad2ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle x\in A}" loading="lazy"></span> gilt: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(f(x))=x}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d04c2f87b1313815142f4089bddf9881977e4bdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.934ex; height:2.843ex;" alt="{\displaystyle f(f(x))=x}" loading="lazy"></span>.

</p><p>Diese Forderung lässt sich auch kompakter formulieren als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ f=\operatorname {id} _{A}}">

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ f=\operatorname {id} _{A}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb781341ab0e1e18bc2c83745e514ba0b88c8c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.255ex; height:2.509ex;" alt="{\displaystyle f\circ f=\operatorname {id} _{A}}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}=\operatorname {id} _{A}}">

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<annotation encoding="application/x-tex">{\displaystyle f^{2}=\operatorname {id} _{A}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89337e1f785a464d2c9e64ced59fcd8d332b41ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.877ex; height:3.009ex;" alt="{\displaystyle f^{2}=\operatorname {id} _{A}}" loading="lazy"></span>.

Dabei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{A}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{A}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f8e9928a7f50859422671451607b974756960fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.404ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{A}}" loading="lazy"></span> die <a href="Identische_Abbildung" title="Identische Abbildung">Identität</a> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>.

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

<ul><li>Jede Involution ist eine <a href="Bijektion" class="mw-redirect" title="Bijektion">Bijektion</a> und es gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}=f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db3c75f82645100e1c08808025d45db94efeeb81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.03ex; height:3.009ex;" alt="{\displaystyle f^{-1}=f}" loading="lazy"></span>.</li>

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<annotation encoding="application/x-tex">{\displaystyle f\colon A\to A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca02cc6a46673db4716f33f5c1eaff9c9e034a0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.413ex; height:2.509ex;" alt="{\displaystyle f\colon A\to A}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon A\to A}">

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<annotation encoding="application/x-tex">{\displaystyle g\colon A\to A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bc06fcb4e31d8f602f4d1168f8ed9bd3035f525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.25ex; height:2.509ex;" alt="{\displaystyle g\colon A\to A}" loading="lazy"></span> Involutionen sind, dann ist ihre <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Komposition</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"></span> genau dann selbst eine Involution, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=g\circ f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49a99c329a3375b7bb3b3074a0383284a5a02401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.277ex; height:2.509ex;" alt="{\displaystyle f\circ g=g\circ f}" loading="lazy"></span> gilt.</li>

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<annotation encoding="application/x-tex">{\displaystyle f\colon A\to A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca02cc6a46673db4716f33f5c1eaff9c9e034a0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.413ex; height:2.509ex;" alt="{\displaystyle f\colon A\to A}" loading="lazy"></span> eine Involution und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon A\to A}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon A\to A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bc06fcb4e31d8f602f4d1168f8ed9bd3035f525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.25ex; height:2.509ex;" alt="{\displaystyle g\colon A\to A}" loading="lazy"></span> eine Bijektion, dann ist die Komposition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f\circ g^{-1}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g\circ f\circ g^{-1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e6c2b63cced739da5060b286cf35c239a6fe420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.235ex; height:3.009ex;" alt="{\displaystyle g\circ f\circ g^{-1}}" loading="lazy"></span> ebenfalls eine Involution. Mit dieser Eigenschaft können neue Involutionen erzeugt werden.</li>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span> eine Bijektion der endlichen Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{n}=\{1,\dotsc ,n\}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{n}=\{1,\dotsc ,n\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0dddd79b1d2fb4c8bba998f3accd3b2d6403ba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.055ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} _{n}=\{1,\dotsc ,n\}}" loading="lazy"></span> (also ein Element der <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen Gruppe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}" loading="lazy"></span></a>), dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span> genau dann involutorisch, wenn es sich als <a href="Produkt_(Mathematik)" title="Produkt (Mathematik)">Produkt</a> aus lauter disjunkten <a href="Vertauschung" class="mw-redirect" title="Vertauschung">Vertauschungen</a> schreiben lässt. Man spricht in diesem Fall von einer <a href="Selbstinverse_Permutation" title="Selbstinverse Permutation">selbstinversen Permutation</a>.</li>

<li>Der <a href="Funktionsgraph" title="Funktionsgraph">Graph</a> einer Involution in den <a href="Reelle_Zahl" title="Reelle Zahl">Reellen Zahlen</a> ist symmetrisch zur <a href="Identische_Abbildung" title="Identische Abbildung">Winkelhalbierenden</a>, die selbst der Graph der trivialen Involution ist. Daraus folgt, dass eine Verschiebung einer Involution entlang der Winkelhalbierenden ebenfalls eine Involution ergibt. Die Involution <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> ist also invariant unter der Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\to f(x-a)+a}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)\to f(x-a)+a}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99b3d735274e9c5c388cc4315dd82d35e029c461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.59ex; height:2.843ex;" alt="{\displaystyle f(x)\to f(x-a)+a}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b044c60e01b54c7116ee355431f37ed846badc53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }" loading="lazy"></span>, dies ist ein Spezialfall der Komposition einer Bijektion mit der Involution und ihrer Inversen.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Involutionen_auf_Vektorräumen"><span id="Involutionen_auf_Vektorr.C3.A4umen"></span>Involutionen auf Vektorräumen</h3></div>

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

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</math></span><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"></span> ein endlichdimensionaler <a href="Vektorraum" title="Vektorraum">Vektorraum</a> über dem Körper <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span>.

</p>

<ul><li>Eine (lineare) <a href="Selbstabbildung" title="Selbstabbildung">Selbstabbildung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \operatorname {End} (V)}">

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<annotation encoding="application/x-tex">{\displaystyle f\in \operatorname {End} (V)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c309fdeccc264f6e60a972bc27d9657dd9b450e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.884ex; height:2.843ex;" alt="{\displaystyle f\in \operatorname {End} (V)}" loading="lazy"></span> ist genau dann involutorisch, wenn das <a href="Minimalpolynom_(Lineare_Algebra)" title="Minimalpolynom (Lineare Algebra)">Minimalpolynom</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> die Form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-1}">

<msup>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c295fc4a24c9128f6ef2659db9f1579090d9d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}-1}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f1a88d34243b98b57c4df9db5724f61b59a4b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x-1}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16df430ed7a23df9b160a5bbd957f306a0c3baa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x+1}" loading="lazy"></span> hat. Das bedeutet insbesondere:

<ul><li>Ist die <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> des Grundkörpers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> von 2 verschieden, so ist jeder involutorische <a href="Endomorphismus" title="Endomorphismus">Endomorphismus</a> <a href="Diagonalisierbar" class="mw-redirect" title="Diagonalisierbar">diagonalisierbar</a> und alle seine <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> liegen in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-1;+1\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49dc7a1f54ceff25df2f6db5d90a633e7cb9554f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle \{-1;+1\}}" loading="lazy"></span>.</li>

<li>Jede Involution <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \operatorname {End} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in \operatorname {End} (V)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c309fdeccc264f6e60a972bc27d9657dd9b450e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.884ex; height:2.843ex;" alt="{\displaystyle f\in \operatorname {End} (V)}" loading="lazy"></span> ist eine <a href="Darstellung_(Gruppe)" title="Darstellung (Gruppe)">Darstellung</a> der Gruppe <a href="Z2_(Gruppe)" title="Z2 (Gruppe)">Z/2Z</a> in der <a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeinen linearen Gruppe</a> GL(V).</li>

<li>Über Körpern <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> mit der Charakteristik 2 gibt es nicht diagonalisierbare involutorische Endomorphismen. So ist im zweidimensionalen Vektorraum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}^{2}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fc44cb159503495ec48bc001f34609525b72215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.475ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{2}^{2}}" loading="lazy"></span> durch die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01b05f8fde8f699d4b1e646e1fbb0c517ca08da4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.82ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}" loading="lazy"></span> eine Involution gegeben, die nicht diagonalisierbar ist.</li></ul></li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="Negatives_und_Kehrwert">Negatives und Kehrwert</h3></div>

<p>Die Abbildungen

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to \mathbb {R} ,\quad x\mapsto -x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af21891396b6492db54192a37ae34e2c985d9bdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.408ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} \to \mathbb {R} ,\quad x\mapsto -x}" loading="lazy"></span></dd></dl>

<p>und

</p>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\times }\to \mathbb {R} ^{\times },\quad x\mapsto {\frac {1}{x}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/877b4a8fef328650cf054e11ed162c519ac8b971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.458ex; height:5.176ex;" alt="{\displaystyle \mathbb {R} ^{\times }\to \mathbb {R} ^{\times },\quad x\mapsto {\frac {1}{x}}}" loading="lazy"></span></dd></dl>

<p>sind Involutionen, denn es gilt

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ad9e29e795867a4a57459bc6301344c00d1f469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.183ex; height:2.843ex;" alt="{\displaystyle -(-x)=x}" loading="lazy"></span> für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9c6d458566aec47a7259762034790c8981aefab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }" loading="lazy"></span></dd></dl>

<p>und

</p>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a915183bfe237557775052c65e442a3e1d0fca6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.919ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{1/x}}=x}" loading="lazy"></span> für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy"></span>.</dd></dl>

<p>Ist allgemein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> eine <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a>, so ist die Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\mapsto -g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d016349b4e532a1ca7ce3d7057de8c6bd688bd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.654ex; height:2.343ex;" alt="{\displaystyle g\mapsto -g}" loading="lazy"></span> (bei additiver Schreibweise) bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\mapsto g^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dbbc406e1f05e3dd468115273949658054a2821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.181ex; height:3.009ex;" alt="{\displaystyle g\mapsto g^{-1}}" loading="lazy"></span> (bei multiplikativer Schreibweise) ein <a href="Gruppenautomorphismus" class="mw-redirect" title="Gruppenautomorphismus">Gruppenautomorphismus</a> und eine Involution. Für eine nichtabelsche Gruppe ist diese Abbildung zwar auch eine Involution, aber kein <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismus</a> (gleichwohl ein <a href="Antihomomorphismus" title="Antihomomorphismus">Gruppen-<i>Anti</i>homomorphismus</a>).

</p>

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

<p>Die Negation in der <a href="Klassische_Logik" title="Klassische Logik">klassischen Logik</a> ist ebenfalls eine Involution, denn es gilt:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f232a9f367d2240aca61301a0e17d33e951d5794.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.806ex; height:2.843ex;" alt="{\displaystyle \forall x(\lnot \lnot x\leftrightarrow x)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Die_komplexe_Konjugation">Die komplexe Konjugation</h3></div>

<p>Beim Rechnen mit <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a> ist das Bilden der <a href="Komplexe_Zahl#Komplexe_Konjugation" title="Komplexe Zahl">konjugiert-komplexen</a> Zahl eine Involution:

Für eine komplexe Zahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+b\mathrm {i} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=a+b\mathrm {i} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d7f54052b27c21d6073ea59a31e499ea689970f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.901ex; height:2.343ex;" alt="{\displaystyle z=a+b\mathrm {i} }" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe0bf0e2324bac843fab916c46d1b3349017a616.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }" loading="lazy"></span> ist die konjugiert-komplexe Zahl

</p>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=z^{*}=a-b\mathrm {i} .}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec2131d123db69711c5c940dc3c39ebd5096b74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.999ex; height:2.509ex;" alt="{\displaystyle {\bar {z}}=z^{*}=a-b\mathrm {i} .}" loading="lazy"></span></dd></dl>

<p>Nochmalige Ausführung der Konjugation liefert

<mover>

</mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\overline {z}}}=z^{**}=a+b\mathrm {i} =z}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a7487bb07224b70c387fb5868dd86dfe34d5eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.387ex; height:3.343ex;" alt="{\displaystyle {\overline {\overline {z}}}=z^{**}=a+b\mathrm {i} =z}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Die_Quaternionen-Konjugation">Die Quaternionen-Konjugation</h3></div>

<p>Zur <a href="Quaternion" title="Quaternion">Quaternion</a>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=x_{0}+x_{1}\mathrm {i} +x_{2}\mathrm {j} +x_{3}\mathrm {k} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eea9a1cf7e845cee5da9af9c600aafe144e63ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.071ex; height:2.509ex;" alt="{\displaystyle x=x_{0}+x_{1}\mathrm {i} +x_{2}\mathrm {j} +x_{3}\mathrm {k} }" loading="lazy"></span></dd></dl>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1},x_{2},x_{3}\in \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79a675d0a98d570624b9e1484ba8e0bd6d0f21f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.156ex; height:2.509ex;" alt="{\displaystyle x_{0},x_{1},x_{2},x_{3}\in \mathbb {R} }" loading="lazy"></span> wird die <a href="Quaternion#Konjugation" title="Quaternion">konjugierte Quaternion</a> durch

</p>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}=x_{0}-x_{1}\mathrm {i} -x_{2}\mathrm {j} -x_{3}\mathrm {k} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd567b90c315b8fd65eae55d76419daf01939c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.071ex; height:2.509ex;" alt="{\displaystyle {\bar {x}}=x_{0}-x_{1}\mathrm {i} -x_{2}\mathrm {j} -x_{3}\mathrm {k} }" loading="lazy"></span></dd></dl>

<p>gebildet. Wegen der Umkehrung der Reihenfolge (wichtig bei nicht-<a href="Kommutativ" class="mw-redirect" title="Kommutativ">kommutativen</a> Ringen!) der Faktoren bei der Multiplikation

</p>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/158f14297cc3d28ee92c5926ba94574083bce8ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.699ex; height:2.676ex;" alt="{\displaystyle {\overline {x\cdot y}}={\bar {y}}\cdot {\bar {x}}}" loading="lazy"></span></dd></dl>

<p>wird diese Konjugation als <a href="Antiautomorphismus" class="mw-redirect" title="Antiautomorphismus">Antiautomorphismus</a> bezeichnet.

</p><p>Nochmalige Ausführung der Konjugation liefert

</p>

<mover>

</mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c031a2dfa8e726dba07dc7f63e7b573abb71decb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.635ex; height:3.176ex;" alt="{\displaystyle {\overline {\overline {x}}}=x.}" loading="lazy"></span></dd></dl>

<p>Sie ist also eine Involution.

</p><p>Beide Eigenschaften zusammen ergeben einen <a href="Antiautomorphismus#Involutiver_Antiautomorphismus" class="mw-redirect" title="Antiautomorphismus">involutiven Antiautomorphismus</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Das_Transponieren_von_Matrizen">Das Transponieren von Matrizen</h3></div>

<p>In der Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{n\times n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1f52ca8e2b5e7ab0a2dcf89f06254a5ef8906f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.247ex; height:2.343ex;" alt="{\displaystyle R^{n\times n}}" loading="lazy"></span> der quadratischen <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrizen</a> über einem Ring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> ist das <a href="Transponierte_Matrix" title="Transponierte Matrix">Transponieren</a>

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot ^{T}\colon R^{n\times n}\rightarrow R^{n\times n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d97c23b1aa3d3500c3d20bf8dd94cabab749551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.179ex; height:2.676ex;" alt="{\displaystyle \cdot ^{T}\colon R^{n\times n}\rightarrow R^{n\times n}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mapsto A^{T}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mapsto A^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50a968f1f3861c355026273a501c61c8de5f8bc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.49ex; height:2.676ex;" alt="{\displaystyle A\mapsto A^{T}}" loading="lazy"></span></dd></dl>

<p>eine Involution. Da <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{n\times n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1f52ca8e2b5e7ab0a2dcf89f06254a5ef8906f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.247ex; height:2.343ex;" alt="{\displaystyle R^{n\times n}}" loading="lazy"></span> ein Ring ist, sogar ein involutiver Antiautomorphismus.

</p><p>Aus dieser Eigenschaft folgt zusammen mit der Selbstinversität der komplexen Konjugation, dass das <a href="Adjungierte_Matrix" title="Adjungierte Matrix">Adjungieren</a> einer Matrix eine Involution ist.

</p>

<div class="mw-heading mw-heading3"><h3 id="Rechnen_in_F2">Rechnen in F<sub>2</sub></h3></div>

<p>In der additiven Gruppe des <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">Restklassenkörpers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fde97f1e971e76227cd0aac645b7b0901d7b668d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}" loading="lazy"></span> ist die Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x+1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/092c91a7a2ccb9481d3da9c5e59cd410c66dd87f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.276ex; height:2.343ex;" alt="{\displaystyle x\mapsto x+1}" loading="lazy"></span> eine Involution:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14995758a82af63a3ea4197d2c31977f54b2f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.22ex; height:2.843ex;" alt="{\displaystyle (x+1)+1=x.}" loading="lazy"></span></dd></dl>

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

<p>In der <a href="Geometrie" title="Geometrie">Geometrie</a> sind Punkt- und Geradenspiegelungen Involutionen.

</p>

<div class="mw-heading mw-heading3"><h3 id="Involutorische_Chiffren">Involutorische Chiffren</h3></div>

<p>Involutorische Chiffren weisen die Eigenart auf, dass der <a href="Algorithmus" title="Algorithmus">Algorithmus</a> zum Verschlüsseln und zum Entschlüsseln identisch ist. Sie sind damit besonders bequem zu handhaben. Ein einfaches Beispiel aus der <a href="Kryptologie" title="Kryptologie">Kryptologie</a> ist die <a href="Verschiebechiffre" class="mw-redirect" title="Verschiebechiffre">Verschiebechiffre</a> <a href="ROT13" title="ROT13">ROT13</a>, bei der zur <a href="Verschl%C3%BCsselung" title="Verschlüsselung">Verschlüsselung</a> jeder Buchstabe durch den um 13 Stellen im <a href="Alphabet_(Kryptologie)" title="Alphabet (Kryptologie)">Alphabet</a> verschobenen Buchstaben ersetzt wird. Die zweimalige Anwendung dieser Methode ergibt eine Verschiebung um 26 Buchstaben und damit wieder den ursprünglichen <a href="Klartext_(Kryptographie)" title="Klartext (Kryptographie)">Klartext</a>. In der Geschichte gab es aber auch wesentlich komplexere involutorische Verschlüsselungsverfahren. Das wohl bekannteste Beispiel ist die deutsche Verschlüsselungsmaschine <a href="Enigma_(Maschine)" title="Enigma (Maschine)">ENIGMA</a>, die im <a href="Zweiter_Weltkrieg" title="Zweiter Weltkrieg">Zweiten Weltkrieg</a> im Nachrichtenverkehr des <a href="Wehrmacht" title="Wehrmacht">deutschen Militärs</a> verwendet wurde.

</p><p>Die logische Funktion <a href="Exklusives_Oder" class="mw-redirect" title="Exklusives Oder">Exklusives Oder</a> ist ebenfalls selbstinvers und wird daher unter anderem in Verschlüsselungsalgorithmen wie <a href="One_Time_Pad" class="mw-redirect" title="One Time Pad">One Time Pad</a> eingesetzt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Körperinvolution"><span id="K.C3.B6rperinvolution"></span>Körperinvolution</h2></div>

<p>Unter einer <b>Körperinvolution</b> versteht man üblicherweise eine Involution, die zugleich ein Körper<a href="Automorphismus" title="Automorphismus">automorphismus</a> ist.

</p><p>Von einer Körperinvolution <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"></span> über einem Körper <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> fordert man also

</p>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\operatorname {id} _{K}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc82ffd77a0d9c85ecc181d8240d5f0d3a2f0208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.116ex; height:3.009ex;" alt="{\displaystyle \sigma ^{2}=\operatorname {id} _{K}}" loading="lazy"></span></dd></dl>

<p>sowie für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b\in K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3ff3a49d65fc590e33a74fd613900dd5924d6ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.168ex; height:2.509ex;" alt="{\displaystyle a,b\in K}" loading="lazy"></span>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (a+b)=\sigma (a)+\sigma (b)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa81822ac4617acca8d65523c8b86bd11ee6750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.651ex; height:2.843ex;" alt="{\displaystyle \sigma (a+b)=\sigma (a)+\sigma (b)}" loading="lazy"></span></dd></dl>

<p>und

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (ab)=\sigma (a)\sigma (b).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f079e661bafb85a1eca8727e7162a915acfb56d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.617ex; height:2.843ex;" alt="{\displaystyle \sigma (ab)=\sigma (a)\sigma (b).}" loading="lazy"></span></dd></dl>

<p>Die bekannteste nichttriviale Körperinvolution ist die <a href="Komplexe_Konjugation" title="Komplexe Konjugation">Konjugation</a> über den <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a>. Aus diesem Grund benutzt man für eine Körperinvolution oft die gleiche Schreibweise wie für die komplexe Konjugation: Anstelle von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (a)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf9ea3f420f9f4df8439a86b1256e4157c632e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.369ex; height:2.843ex;" alt="{\displaystyle \sigma (a)}" loading="lazy"></span> wird häufig <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {a}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/032e261791bd07a59cf1419352fc66f7901d4b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.345ex; height:2.343ex;" alt="{\displaystyle {\overline {a}}}" loading="lazy"></span> geschrieben.

</p><p>Ein anderes Beispiel ist der Automorphismus des Körpers

</p>

</mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \left({\sqrt {2}}\right)=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d403b80a99947adbc10e17130eb6936846d0ff44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.28ex; height:3.343ex;" alt="{\displaystyle \mathbb {Q} \left({\sqrt {2}}\right)=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},}" loading="lazy"></span></dd></dl>

<p>der durch

</p>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {2}}\mapsto a-b{\sqrt {2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61cbe51b2d7dcfe8ce375ac7e6a9fdf4be2d65bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.946ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {2}}\mapsto a-b{\sqrt {2}}}" loading="lazy"></span></dd></dl>

<p>definiert ist. Man beachte, dass er im Unterschied zur komplexen Konjugation den Betrag nicht erhält:

</p>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |7-5{\sqrt {2}}|\approx 0{,}1,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/720b15ec7422e81d2f7fc9b3e41898e9b7eb2097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.274ex; height:3.176ex;" alt="{\displaystyle |7-5{\sqrt {2}}|\approx 0{,}1,}" loading="lazy"></span> aber <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |7+5{\sqrt {2}}|\approx 14{,}1.}">

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |7+5{\sqrt {2}}|\approx 14{,}1.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68f5cfa32ef3af10a8a7826dea8771f4a6856f39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.437ex; height:3.176ex;" alt="{\displaystyle |7+5{\sqrt {2}}|\approx 14{,}1.}" loading="lazy"></span></dd></dl>

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

<ul><li><cite class="lang" lang="en" dir="auto" style="font-style:italic">Involution</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/involution">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Involution+%28Mathematik%29&rft.atitle=Involution&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

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