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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Einheitsmatrix</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>Die <b>Einheitsmatrix</b> oder <b>Identitätsmatrix</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine <a href="Quadratische_Matrix" class="mw-redirect" title="Quadratische Matrix">quadratische Matrix</a>, deren Elemente auf der <a href="Hauptdiagonale" title="Hauptdiagonale">Hauptdiagonale</a> eins und überall sonst null sind. Die Einheitsmatrix ist im <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</a> der quadratischen Matrizen das <a href="Neutrales_Element" title="Neutrales Element">neutrale Element</a> bezüglich der <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikation</a>. Sie ist <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a>, <a href="Involution_(Mathematik)" title="Involution (Mathematik)">selbstinvers</a>, <a href="Idempotenz" title="Idempotenz">idempotent</a> und hat maximalen <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a>. Die Einheitsmatrix ist die <a href="Darstellungsmatrix" class="mw-redirect" title="Darstellungsmatrix">Darstellungsmatrix</a> der <a href="Identische_Abbildung" title="Identische Abbildung">Identitätsabbildung</a> eines endlichdimensionalen <a href="Vektorraum" title="Vektorraum">Vektorraums</a>. Sie wird unter anderem bei der Definition des <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristischen Polynoms</a> einer Matrix, <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonaler</a> und <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitärer</a> Matrizen, sowie in einer Reihe geometrischer Abbildungen verwendet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</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> ein Ring mit <a href="Nullelement" class="mw-redirect" title="Nullelement">Nullelement</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 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> und <a href="Einselement" class="mw-redirect" title="Einselement">Einselement</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 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>, dann ist die Einheitsmatrix <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 I_{n}\in R^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}\in R^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8b64e15611017d1234fadba4b990f957041b6e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.329ex; height:2.676ex;" alt="{\displaystyle I_{n}\in R^{n\times n}}" loading="lazy"></span> die <a href="Quadratische_Matrix" class="mw-redirect" title="Quadratische Matrix">quadratische Matrix</a>
</p>
<dl><dd><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 I_{n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&1\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mo>⋱<!-- ⋱ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋱<!-- ⋱ --></mo>
</mtd>
<mtd>
<mo>⋱<!-- ⋱ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&1\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535e3e4b3399a01ec4666e7d99a6765a2431f89c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:25.408ex; height:15.843ex;" alt="{\displaystyle I_{n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&1\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Eine Einheitsmatrix ist demnach eine <a href="Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrix</a>, bei der alle Elemente auf der <a href="Hauptdiagonale" title="Hauptdiagonale">Hauptdiagonale</a> gleich <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> sind. Als Schreibweise ist neben <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 I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}}</annotation>
</semantics>
</math></span><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"></span> (von <i>Identität</i>) auch <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_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy"></span> (von <i>Einheit</i>) gebräuchlich. Falls die Dimension aus dem Zusammenhang hervorgeht, wird auch häufig auf den Index <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span> verzichtet und nur <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 I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
</semantics>
</math></span><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"></span> beziehungsweise <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> geschrieben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</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> der <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> der <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> und bezeichnen <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 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" 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 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> die Zahlen <a href="Null" title="Null">Null</a> und <a href="Eins" title="Eins">Eins</a>, so sind Beispiele für Einheitsmatrizen:
</p>
<dl><dd><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 I_{1}={\begin{pmatrix}1\end{pmatrix}},\ I_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\ I_{3}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},\ I_{4}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mtext> </mtext>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
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<mn>1</mn>
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<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mtext> </mtext>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mtext> </mtext>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{1}={\begin{pmatrix}1\end{pmatrix}},\ I_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\ I_{3}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},\ I_{4}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f28b4121eb727c2312e0f53718a1314befe3d826.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:67.477ex; height:12.509ex;" alt="{\displaystyle I_{1}={\begin{pmatrix}1\end{pmatrix}},\ I_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\ I_{3}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},\ I_{4}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Elemente">Elemente</h3></div>
<p>Die Elemente einer Einheitsmatrix lassen sich mit dem <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a>
</p>
<dl><dd><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 \delta _{ij}=\left\{{\begin{matrix}1\quad {\text{falls}}\quad i=j\\0\quad {\text{falls}}\quad i\neq j\end{matrix}}\right.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
<mspace width="1em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls</mtext>
</mrow>
<mspace width="1em"></mspace>
<mi>i</mi>
<mo>=</mo>
<mi>j</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
<mspace width="1em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls</mtext>
</mrow>
<mspace width="1em"></mspace>
<mi>i</mi>
<mo>≠<!-- ≠ --></mo>
<mi>j</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}=\left\{{\begin{matrix}1\quad {\text{falls}}\quad i=j\\0\quad {\text{falls}}\quad i\neq j\end{matrix}}\right.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a222073a19d007c0c8900120d800fc8977f5fd8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.853ex; height:6.176ex;" alt="{\displaystyle \delta _{ij}=\left\{{\begin{matrix}1\quad {\text{falls}}\quad i=j\\0\quad {\text{falls}}\quad i\neq j\end{matrix}}\right.}" loading="lazy"></span></dd></dl>
<p>angeben. Die Einheitsmatrix der Größe <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./_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"></span> kann so einfach durch
</p>
<dl><dd><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 I_{n}=(\delta _{ij})_{i,j\in \{1,\ldots ,n\}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
<mo fence="false" stretchy="false">}</mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}=(\delta _{ij})_{i,j\in \{1,\ldots ,n\}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e5de96eb9569d5e6a3c177c975aedb86ee7e6ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.982ex; height:3.176ex;" alt="{\displaystyle I_{n}=(\delta _{ij})_{i,j\in \{1,\ldots ,n\}}}" loading="lazy"></span></dd></dl>
<p>notiert werden. Die Zeilen und Spalten der Einheitsmatrix sind die kanonischen <a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektoren</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_{1},\ldots ,e_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}" loading="lazy"></span>, und man schreibt entsprechend
</p>
<dl><dd><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 I_{n}=(e_{1},\ldots ,e_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}=(e_{1},\ldots ,e_{n})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11589d0b9c601a8aae9292d89152d72f39248071.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.767ex; height:2.843ex;" alt="{\displaystyle I_{n}=(e_{1},\ldots ,e_{n})}" loading="lazy"></span>,</dd></dl>
<p>wenn die Einheitsvektoren Spaltenvektoren sind.
</p>
<div class="mw-heading mw-heading3"><h3 id="Neutralität"><span id="Neutralit.C3.A4t"></span>Neutralität</h3></div>
<p>Für jede 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 A\in R^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in R^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b86f150f527b6ae03f33869a6a05fa139c943d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.288ex; height:2.343ex;" alt="{\displaystyle A\in R^{m\times n}}" loading="lazy"></span> gilt
</p>
<dl><dd><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 I_{m}\cdot A=A\cdot I_{n}=A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi>A</mi>
<mo>=</mo>
<mi>A</mi>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{m}\cdot A=A\cdot I_{n}=A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f73e8a2b0830ed6164ea36d96e4ab174fb29246.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.724ex; height:2.509ex;" alt="{\displaystyle I_{m}\cdot A=A\cdot I_{n}=A}" loading="lazy"></span>.</dd></dl>
<p>Demnach ergibt das Produkt aus einer beliebigen Matrix mit der Einheitsmatrix wieder die gleiche Matrix. Die Menge der quadratischen Matrizen bildet zusammen mit der <a href="Matrizenaddition" title="Matrizenaddition">Matrizenaddition</a> und der <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikation</a> einen (nichtkommutativen) 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^{n\times n},+,\cdot )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
<mo>,</mo>
<mo>+</mo>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (R^{n\times n},+,\cdot )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f968f76d9c1ddffab7f38fde4f515c1885c4d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.579ex; height:2.843ex;" alt="{\displaystyle (R^{n\times n},+,\cdot )}" loading="lazy"></span>. Die Einheitsmatrix ist dann das <a href="Einselement" class="mw-redirect" title="Einselement">Einselement</a> in diesem <a href="Matrizenring" title="Matrizenring">Matrizenring</a>, also das <a href="Neutrales_Element" title="Neutrales Element">neutrale Element</a> bezüglich der Matrizenmultiplikation.
</p>
<div class="mw-heading mw-heading3"><h3 id="Symmetrien">Symmetrien</h3></div>
<p>Die Einheitsmatrix ist <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a>, das heißt für ihre <a href="Transponierte_Matrix" title="Transponierte Matrix">Transponierte</a> gilt
</p>
<dl><dd><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 (I_{n})^{T}=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (I_{n})^{T}=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/811d180d3726fd86c841990eb2150bbc5b1b3f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.78ex; height:3.176ex;" alt="{\displaystyle (I_{n})^{T}=I_{n}}" loading="lazy"></span>,</dd></dl>
<p>und <a href="Involution_(Mathematik)" title="Involution (Mathematik)">selbstinvers</a>, das heißt für ihre <a href="Inverse_Matrix" title="Inverse Matrix">Inverse</a> gilt ebenfalls
</p>
<dl><dd><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 (I_{n})^{-1}=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (I_{n})^{-1}=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6817bb4dae0f7e577cc98c8032264ddb740a5b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.724ex; height:3.176ex;" alt="{\displaystyle (I_{n})^{-1}=I_{n}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Kenngrößen"><span id="Kenngr.C3.B6.C3.9Fen"></span>Kenngrößen</h3></div>
<p>Für die <a href="Determinante" title="Determinante">Determinante</a> der Einheitsmatrix gilt
</p>
<dl><dd><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 {det} (I_{n})=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>det</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (I_{n})=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eff882d53134e2190b9035bf986f587ea891644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.541ex; height:2.843ex;" alt="{\displaystyle \operatorname {det} (I_{n})=1}" loading="lazy"></span>,</dd></dl>
<p>was eine der drei <a href="Determinante#Axiomatische_Beschreibung" title="Determinante">definierenden Eigenschaften</a> einer Determinante ist. Für die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> der Einheitsmatrix gilt
</p>
<dl><dd><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 {spur} (I_{n})=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {spur} (I_{n})=n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33c312db6cd33698c70a0c2a2ee8ec26aa26f497.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.957ex; height:2.843ex;" alt="{\displaystyle \operatorname {spur} (I_{n})=n}" loading="lazy"></span>.</dd></dl>
<p>Handelt es sich bei dem Ring um <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 {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" 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 \mathbb {Q} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" 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 \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" 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 \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span>, erhält man demnach <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 {spur} (I_{n})=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {spur} (I_{n})=n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33c312db6cd33698c70a0c2a2ee8ec26aa26f497.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.957ex; height:2.843ex;" alt="{\displaystyle \operatorname {spur} (I_{n})=n}" loading="lazy"></span>. Das <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristische Polynom</a> der Einheitsmatrix ergibt sich als
</p>
<dl><dd><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 \chi _{I_{n}}(\lambda )=(\lambda -1)^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi _{I_{n}}(\lambda )=(\lambda -1)^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7b222f97b0e17d32a36329faecd17594668e1eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.025ex; height:2.843ex;" alt="{\displaystyle \chi _{I_{n}}(\lambda )=(\lambda -1)^{n}}" loading="lazy"></span>.</dd></dl>
<p>Der einzige <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> ist demnach <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 \lambda =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda =1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/543b4490416437b7c80ea473bbcac0e4ab7a7f11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =1}" loading="lazy"></span> mit Vielfachheit <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span>. In der Tat 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 I_{n}\cdot x=1\cdot x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>⋅<!-- ⋅ --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}\cdot x=1\cdot x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1454d2c0c30e9b0899dbdbf5d98033d994978789.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.52ex; height:2.509ex;" alt="{\displaystyle I_{n}\cdot x=1\cdot 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}">
<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_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.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"></span> des <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduls</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 R^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2ebce0f18894435d56a8fe182e22f135aa8ed07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.982ex; height:2.343ex;" alt="{\displaystyle R^{n}}" loading="lazy"></span>. 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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</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> ein <a href="Kommutativer_Ring" class="mw-redirect" title="Kommutativer Ring">kommutativer Ring</a>, so ist der <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a> der Einheitsmatrix durch
</p>
<dl><dd><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 {rang} (I_{n})=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>rang</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {rang} (I_{n})=n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/166eb3b0d26553eb594489ed72964df165bac563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.073ex; height:2.843ex;" alt="{\displaystyle \operatorname {rang} (I_{n})=n}" loading="lazy"></span></dd></dl>
<p>gegeben.
</p>
<div class="mw-heading mw-heading3"><h3 id="Potenzen">Potenzen</h3></div>
<p>Die Einheitsmatrix ist <a href="Idempotenz" title="Idempotenz">idempotent</a>, das heißt
</p>
<dl><dd><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 I_{n}\cdot I_{n}=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{n}\cdot I_{n}=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff878f052ea5b5a7a2a0032bc72625fb7d4b3fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.502ex; height:2.509ex;" alt="{\displaystyle I_{n}\cdot I_{n}=I_{n}}" loading="lazy"></span>,</dd></dl>
<p>und sie ist die einzige Matrix mit vollem Rang mit dieser Eigenschaft. Für das <a href="Matrixexponential" title="Matrixexponential">Matrixexponential</a> einer reellen oder komplexen Einheitsmatrix gilt damit
</p>
<dl><dd><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 \exp(I_{n})=\sum _{k=0}^{\infty }{\frac {(I_{n})^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\cdot I_{n}=e\cdot I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>e</mi>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \exp(I_{n})=\sum _{k=0}^{\infty }{\frac {(I_{n})^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\cdot I_{n}=e\cdot I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f84e6e1efcf72e7067001c63eeb58d10361a29b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.978ex; height:7.176ex;" alt="{\displaystyle \exp(I_{n})=\sum _{k=0}^{\infty }{\frac {(I_{n})^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\cdot I_{n}=e\cdot I_{n}}" loading="lazy"></span>,</dd></dl>
<p>wobei <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/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> die <a href="Eulersche_Zahl" title="Eulersche Zahl">eulersche Zahl</a> ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwendung">Verwendung</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Lineare_Algebra">Lineare Algebra</h3></div>
<p>Die Menge der regulären Matrizen der Größe <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./_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"></span> bildet mit der Matrizenmultiplikation die <a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeine lineare Gruppe</a>. Für alle Matrizen <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</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> dieser <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> und ihre <a href="Inverse_Matrix" title="Inverse Matrix">Inversen</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^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A^{-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ba3a7118652cffd5de466dc439ee9184371d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.076ex; height:2.676ex;" alt="{\displaystyle A^{-1}}" loading="lazy"></span> gilt dann
</p>
<dl><dd><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^{-1}\cdot A=A\cdot A^{-1}=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>A</mi>
<mo>=</mo>
<mi>A</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A^{-1}\cdot A=A\cdot A^{-1}=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04cd7a6422cdb2768596d90054dc9c4540ce7a1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.435ex; height:3.009ex;" alt="{\displaystyle A^{-1}\cdot A=A\cdot A^{-1}=I_{n}}" loading="lazy"></span>.</dd></dl>
<p>Das <a href="Zentrum_(Algebra)" title="Zentrum (Algebra)">Zentrum</a> dieser Gruppe sind gerade die <a href="Vielfaches" title="Vielfaches">Vielfachen</a> (ungleich null) der Einheitsmatrix. Für eine <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrix</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\in \mathbb {R} ^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in \mathbb {R} ^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebb63eca276c71633e0939f5c5d4ef4bfc565518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.745ex; height:2.343ex;" alt="{\displaystyle A\in \mathbb {R} ^{n\times n}}" loading="lazy"></span> gilt nach Definition
</p>
<dl><dd><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\cdot A^{T}=A^{T}\cdot A=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>A</mi>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\cdot A^{T}=A^{T}\cdot A=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51a7cea5e3f84e8be7745d64a543988241f06d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.548ex; height:3.009ex;" alt="{\displaystyle A\cdot A^{T}=A^{T}\cdot A=I_{n}}" loading="lazy"></span></dd></dl>
<p>und entsprechend dazu für eine <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitäre Matrix</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\in \mathbb {C} ^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in \mathbb {C} ^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b3023de9ab6f59e511d7c8ae72d03b64bcecbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.745ex; height:2.343ex;" alt="{\displaystyle A\in \mathbb {C} ^{n\times n}}" loading="lazy"></span>
</p>
<dl><dd><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\cdot A^{H}=A^{H}\cdot A=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>H</mi>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>H</mi>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>A</mi>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\cdot A^{H}=A^{H}\cdot A=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e643f0c2078cb4027a251aeef320278b216267e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.152ex; height:3.009ex;" alt="{\displaystyle A\cdot A^{H}=A^{H}\cdot A=I_{n}}" loading="lazy"></span>.</dd></dl>
<p>Diese Matrizen bilden jeweils <a href="Untergruppe" title="Untergruppe">Untergruppen</a> der entsprechenden allgemeinen linearen Gruppe. Die nullte <a href="Matrixpotenz" title="Matrixpotenz">Potenz</a> einer quadratischen 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 A\in R^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in R^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c92c05e1106c48e72897dabd111e38a742d477f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.831ex; height:2.343ex;" alt="{\displaystyle A\in R^{n\times n}}" loading="lazy"></span> wird als
</p>
<dl><dd><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^{0}=I_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A^{0}=I_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c902c68a4d29a5ff5d740cc91cad565c02af8040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.137ex; height:3.009ex;" alt="{\displaystyle A^{0}=I_{n}}" loading="lazy"></span></dd></dl>
<p>festgelegt. Weiter wird die Einheitsmatrix bei der Definition des <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristischen Polynoms</a>
</p>
<dl><dd><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 \chi _{A}(\lambda )=\det(A-\lambda I_{n}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>−<!-- − --></mo>
<mi>λ<!-- λ --></mi>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(\lambda )=\det(A-\lambda I_{n}).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c8738d53d2b0f254b08d9130c97dfdd35ebbed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.049ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(\lambda )=\det(A-\lambda I_{n}).}" loading="lazy"></span></dd></dl>
<p>einer quadratischen Matrix verwendet. Die Einheitsmatrix ist die <a href="Darstellungsmatrix" class="mw-redirect" title="Darstellungsmatrix">Darstellungsmatrix</a> der <a href="Identische_Abbildung" title="Identische Abbildung">Identitätsabbildung</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 \operatorname {id} \colon V\to V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>id</mi>
<mo>:<!-- : --></mo>
<mi>V</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {id} \colon V\to V}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69bf682b0aabc4f09ff1b683c0dd9519e943402a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.162ex; height:2.176ex;" alt="{\displaystyle \operatorname {id} \colon V\to V}" loading="lazy"></span> eines endlichdimensionalen <a href="Vektorraum" title="Vektorraum">Vektorraums</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 V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./_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>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Geometrie">Geometrie</h3></div>
<p>In der <a href="Analytische_Geometrie" title="Analytische Geometrie">analytischen Geometrie</a> werden Einheitsmatrizen unter anderem bei der Definition folgender Abbildungsmatrizen <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 T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> verwendet:
</p>
<ul><li><a href="Punktspiegelung" class="mw-redirect" title="Punktspiegelung">Punktspiegelung</a> am <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a>:<br> <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 T=-I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T=-I}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85b5ce5c845ac3785661f0d8f466d123167baa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.715ex; height:2.343ex;" alt="{\displaystyle T=-I}" loading="lazy"></span></li>
<li><a href="Zentrische_Streckung" title="Zentrische Streckung">Zentrische Streckung</a> mit dem Streckungsfaktor <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 m>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m>0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/501173910e6da8425b4e9d44a4e8643620bc2464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>0}" loading="lazy"></span> und dem Ursprung als Zentrum:<br> <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 T=m\cdot I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mi>m</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T=m\cdot I}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d28a44b48806ede6911acdb650fbc497b7c5f0f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.626ex; height:2.176ex;" alt="{\displaystyle T=m\cdot I}" loading="lazy"></span></li>
<li><a href="Spiegelung_(Geometrie)" title="Spiegelung (Geometrie)">Spiegelung</a> an einer <a href="Ursprungsgerade" title="Ursprungsgerade">Ursprungsgerade</a> mit <a href="Richtungsvektor" class="mw-redirect" title="Richtungsvektor">Einheits-Richtungsvektor</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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span>:<br> <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 T=2vv^{T}-I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mn>2</mn>
<mi>v</mi>
<msup>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T=2vv^{T}-I}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/273299d90515a6f14fd8a93ffdee7ee9c9c6ebc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.554ex; height:2.843ex;" alt="{\displaystyle T=2vv^{T}-I}" loading="lazy"></span></li>
<li><a href="Spiegelung_(Geometrie)" title="Spiegelung (Geometrie)">Spiegelung</a> an einer Ursprungsgerade (2D) oder <a href="Ursprungsebene" title="Ursprungsebene">Ursprungsebene</a> (3D) mit <a href="Normalenvektor" title="Normalenvektor">Einheits-Normalenvektor</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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span>:<br> <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 T=I-2nn^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>n</mi>
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T=I-2nn^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf4d9f3fa22b2d259899e60fe0e6d1e568c02a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.088ex; height:2.843ex;" alt="{\displaystyle T=I-2nn^{T}}" loading="lazy"></span></li>
<li><a href="Projektion_(lineare_Algebra)" title="Projektion (lineare Algebra)">Projektion</a> auf den <a href="Komplement%C3%A4rraum" title="Komplementärraum">Komplementärraum</a>, 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 P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> eine Projektionsmatrix auf eine Ursprungsebene oder -gerade ist:<br> <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 T=I-P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T=I-P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09f79ed42cbbb26938bd2f276a526a59f6d2941f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.492ex; height:2.343ex;" alt="{\displaystyle T=I-P}" loading="lazy"></span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Programmierung">Programmierung</h2></div>
<p>In dem numerischen Softwarepaket <a href="MATLAB" class="mw-redirect" title="MATLAB">MATLAB</a> wird die Einheitsmatrix der Größe <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./_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"></span> durch die Funktion <code>eye(n)</code> erzeugt.<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> In <a href="Mathematica" title="Mathematica">Mathematica</a> erhält man die Einheitsmatrix durch <code>IdentityMatrix[n]</code>. In der Python-Bibliothek numpy wird eine Einheitsmatrix durch <code>numpy.identity(n)</code> erzeugt. Die Funktion <code>numpy.eye(N, M, k)</code> mit N=Zeilen, M=Spalten und k=Position der Hauptdiagonalen, bietet eine allgemeinere Möglichkeit, Diagonalarrays, mit Einsen zu erzeugen.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Einsmatrix" title="Einsmatrix">Einsmatrix</a>, eine Matrix, die nur aus Einsen besteht</li>
<li><a href="Nullmatrix" title="Nullmatrix">Nullmatrix</a>, eine Matrix, die nur aus Nullen besteht</li>
<li><a href="Standardmatrix" title="Standardmatrix">Standardmatrix</a>, eine Matrix, die aus genau einer Eins und sonst nur Nullen besteht</li>
<li><a href="Permutationsmatrix" title="Permutationsmatrix">Permutationsmatrix</a>, eine Matrix, die durch Zeilen- oder Spaltenvertauschungen aus einer Einheitsmatrix entsteht</li>
<li><a href="Elementarmatrix" title="Elementarmatrix">Elementarmatrix</a>, eine Matrix, die sich nur an einer Position oder durch Zeilentausch von einer Einheitsmatrix unterscheidet</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: <cite style="font-style:italic">Lineare Algebra</cite>. Springer, 2006, ISBN 3-540-29884-3.<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:Einheitsmatrix&rft.au=Siegfried+Bosch&rft.btitle=Lineare+Algebra&rft.date=2006&rft.genre=book&rft.isbn=3540298843&rft.pub=Springer" style="display:none"> </span></li>
<li>Karsten Schmidt, Götz Trenkler: <cite style="font-style:italic">Einführung in die Moderne Matrix-Algebra</cite>. Springer, 2006, ISBN 3-540-33008-9.<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:Einheitsmatrix&rft.au=Karsten+Schmidt%2C+G%C3%B6tz+Trenkler&rft.btitle=Einf%C3%BChrung+in+die+Moderne+Matrix-Algebra&rft.date=2006&rft.genre=book&rft.isbn=3540330089&rft.pub=Springer" style="display:none"> </span></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/IdentityMatrix.html"><i>Identity Matrix</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li>mathcam: <a rel="nofollow" class="external text" href="https://planetmath.org/identitymatrix"><i>Identity matrix</i>.</a> In: <i><a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</i> (englisch)</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Christoph W. Überhuber, Stefan Katzenbeisser, Dirk Praetorius: <cite style="font-style:italic">MATLAB 7: Eine Einführung</cite>. Springer, 2007, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>18</span>.<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:Einheitsmatrix&rft.au=Christoph+W.+%C3%9Cberhuber%2C+Stefan+Katzenbeisser%2C+Dirk+Praetorius&rft.btitle=MATLAB+7%3A+Eine+Einf%C3%BChrung&rft.date=2007&rft.genre=book&rft.pages=18&rft.pub=Springer" style="display:none"> </span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Bernd Klein: <cite style="font-style:italic">Numerisches Python: Arbeiten mit NumPy, Matplotlib und Pandas</cite>. Hanser, München 2019, ISBN 978-3-446-45076-9, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>67</span>.<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:Einheitsmatrix&rft.au=Bernd+Klein&rft.btitle=Numerisches+Python%3A+Arbeiten+mit+NumPy%2C+Matplotlib+und+Pandas&rft.date=2019&rft.genre=book&rft.isbn=9783446450769&rft.pages=67&rft.place=M%C3%BCnchen&rft.pub=Hanser" style="display:none"> </span></span>
</li>
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