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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Endomorphisme</span></h1>
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<p>En <a href="Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, un <b>endomorphisme</b> est un <a href="Morphisme" title="Morphisme">morphisme</a> (ou homomorphisme) d'un objet mathématique dans lui-même. Ainsi, un endomorphisme d'<a href="Espace_vectoriel" title="Espace vectoriel">espace vectoriel</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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> est une <a href="Application_lin%C3%A9aire" title="Application linéaire">application linéaire</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:E\to E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:E\to E}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5734d21d7824c214d411509d2cbecf2872358571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.381ex; height:2.509ex;" alt="{\displaystyle f:E\to E}" loading="lazy"></span>, et un endomorphisme de <a href="Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</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 G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</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> est un <a href="Morphisme_de_groupes" title="Morphisme de groupes">morphisme de groupes</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:G\to G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>G</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:G\to G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17f1d5bde362fa99070774ba0f19a6bb93000ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.483ex; height:2.509ex;" alt="{\displaystyle f:G\to G}" loading="lazy"></span>, etc. En général, nous pouvons parler d'endomorphisme de n'importe quelle <a href="Th%C3%A9orie_des_cat%C3%A9gories" title="Théorie des catégories">catégorie</a>.
</p><p>Étant donné un objet <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><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"></span> d'une catégorie <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 C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> et deux endomorphismes <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_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> et <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> de <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><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"></span> (donc de type <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\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83d8a6029587ee9b365bdeab1e2f4b7c469b0219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}" loading="lazy"></span>), la <a href="Composition_de_fonctions" title="Composition de fonctions">composée</a> de <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> par <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_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>, notée <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation>
</semantics>
</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> (prononcer <i>f rond g</i>), est aussi un endomorphisme de <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><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"></span> (elle a aussi le type <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\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83d8a6029587ee9b365bdeab1e2f4b7c469b0219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}" loading="lazy"></span>). Comme l'<a href="Application_identit%C3%A9" title="Application identité">application identité</a> de <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><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"></span> est aussi un endomorphisme de <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><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"></span>, nous voyons que l'ensemble de tous les endomorphismes de <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><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"></span> forme un <a href="Mono%C3%AFde" title="Monoïde">monoïde</a>, noté <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 {End} _{C}(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>End</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {End} _{C}(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1527e0738c536a2d9ef907b92ca70d1f97ae7dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.438ex; height:2.843ex;" alt="{\displaystyle \operatorname {End} _{C}(X)}" loading="lazy"></span> ou simplement <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 {End} (X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>End</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {End} (X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f0566348bfc0a832aa7a995f7304fbc0edb7d6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.957ex; height:2.843ex;" alt="{\displaystyle \operatorname {End} (X)}" loading="lazy"></span>, si la catégorie est connue.
</p>

<div class="mw-heading mw-heading2"><h2 id="Anneau_des_endomorphismes">Anneau des endomorphismes</h2></div>
<p>Dans de nombreuses situations, il est possible d'additionner les endomorphismes, et avec la composition des applications, les endomorphismes d'un objet donné forment un <a href="Anneau_unitaire" title="Anneau unitaire">anneau</a>, appelé l'anneau des endomorphismes&nbsp;<a href="https://en.wikipedia.org/wiki/Endomorphism_ring" class="extiw external" title="en:Endomorphism ring"><span class="indicateur-langue" title="Article en anglais&nbsp;: «&nbsp;Endomorphism ring&nbsp;»">(en)</span></a> de l'objet. Cela est possible, par exemple, dans les catégories des <a href="Groupe_ab%C3%A9lien" title="Groupe abélien">groupes abéliens</a>, des <a href="Module_sur_un_anneau" title="Module sur un anneau">modules</a>, des <a href="Espace_vectoriel" title="Espace vectoriel">espaces vectoriels</a>, et plus généralement dans toutes les catégories préadditives&nbsp;<a href="https://en.wikipedia.org/wiki/Preadditive_category" class="extiw external" title="en:Preadditive category"><span class="indicateur-langue" title="Article en anglais&nbsp;: «&nbsp;Preadditive category&nbsp;»">(en)</span></a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Automorphismes">Automorphismes</h2></div>
<p>Un <a href="Isomorphisme" title="Isomorphisme">isomorphisme</a> est un morphisme qui possède un morphisme réciproque, on peut dire que c'est un morphisme <a href="Bijection" title="Bijection">bijectif</a>.
</p><p>Un endomorphisme qui est aussi un isomorphisme est appelé un <a href="Automorphisme" title="Automorphisme">automorphisme</a>.
</p><p>On a donc les implications suivantes&nbsp;:
</p>
<table border="0" align="center">

<tbody><tr>
<td align="center" width="42%">automorphisme
</td>
<td align="center" width="16%">⇒
</td>
<td align="center" width="42%">isomorphisme
</td></tr>
<tr>
<td align="center">⇓
</td>
<td>
</td>
<td align="center">⇓
</td></tr>
<tr>
<td align="center">endomorphisme
</td>
<td align="center">⇒
</td>
<td align="center">(homo)morphisme
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="Endomorphismes_linéaires"><span id="Endomorphismes_lin.C3.A9aires"></span>Endomorphismes linéaires</h2></div>

<p>Un <a href="Endomorphisme_lin%C3%A9aire" title="Endomorphisme linéaire">endomorphisme linéaire</a> est un endomorphisme d'<a href="Espace_vectoriel" title="Espace vectoriel">espaces vectoriels</a>. Ils permettent de décrire les transformations linéaires comme les <a href="Rotation_vectorielle" title="Rotation vectorielle">rotations</a>, les <a href="Homoth%C3%A9tie" title="Homothétie">homothéties</a> ou les <a href="Projection_orthogonale" title="Projection orthogonale">projections</a>.
</p><p>En dimension finie, un endomorphisme linéaire <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:V\to V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>V</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:V\to V}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df0954aff002f457e0e801e01d1154df6b1a4f5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.404ex; height:2.509ex;" alt="{\displaystyle f:V\to V}" loading="lazy"></span> est uniquement déterminé par l'image de sa <a href="Base_(alg%C3%A8bre_lin%C3%A9aire)" title="Base (algèbre linéaire)">base</a>. Un endomorphisme peut donc s'écrire comme une <a href="Matrice_d'une_application_lin%C3%A9aire" title="Matrice d'une application linéaire">matrice</a> carrée<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>.
</p><p>La <a href="R%C3%A9duction_d'endomorphisme" title="Réduction d'endomorphisme">réduction d'endomorphisme</a> consiste à chercher une base dans laquelle la matrice associé est facilement manipulable.
</p>
<div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2></div>
<ul><li><a href="Morphisme" title="Morphisme">Morphisme</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2></div>
<div style="font-size:85%; padding-left:1.6em; margin:0.3em 0;"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="">«&nbsp;<a class="external text" href="https://en.wikipedia.org/wiki/Endomorphism?oldid=7846829">Endomorphism</a>&nbsp;» <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Endomorphism?action=history">voir la liste des auteurs</a>)</small></span>.</div>
<ul><li><span class="ouvrage" id="2002"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <cite style="font-style:normal" lang="en">«&nbsp;Endomorphisme&nbsp;»</cite>, dans <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a>, <cite class="italique" lang="en"><a href="Encyclop%C3%A6dia_of_Mathematics" title="Encyclopædia of Mathematics">Encyclopædia of Mathematics</a></cite>, <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <time>2002</time> <small style="line-height:1em;">(<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<span class="nowrap">978-1556080104</span>, <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/e035600">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Encyclop%C3%A6dia+of+Mathematics&amp;rft.atitle=Endomorphisme&amp;rft.pub=Springer&amp;rft.date=2002&amp;rft.isbn=978-1556080104&amp;rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEndomorphisme"></span></span></li></ul>
<div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Grifone2011"><span class="ouvrage" id="Joseph_Grifone2011">Joseph <span class="nom_auteur">Grifone</span>, <cite class="italique">Algèbre linéaire</cite>, Cépaduès éd, <time>2011</time> <small style="line-height:1em;">(<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<span class="nowrap">978-2-85428-962-6</span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Alg%C3%A8bre+lin%C3%A9aire&amp;rft.pub=C%C3%A9padu%C3%A8s+%C3%A9d&amp;rft.aulast=Grifone&amp;rft.aufirst=Joseph&amp;rft.date=2011&amp;rft.isbn=978-2-85428-962-6&amp;rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEndomorphisme"></span></span></span></span>
</li>
</ol></div>
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