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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Standardmatrix</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>Eine <b>Standardmatrix</b>, <b>Standard-Einheitsmatrix</b> oder <b>Matrixeinheit</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>, bei der genau ein Eintrag eins ist und alle anderen Einträge null sind. Jede Standardmatrix lässt sich als <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadisches Produkt</a> von kanonischen <a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektoren</a> darstellen. Die Menge der Standardmatrizen bildet die <a href="Standardbasis" title="Standardbasis">Standardbasis</a> für den <a href="Matrizenraum" title="Matrizenraum">Matrizenraum</a>. Sie werden unter anderem zur Definition von <a href="Elementarmatrix" title="Elementarmatrix">Elementarmatrizen</a> verwendet, die beim <a href="Gau%C3%9Fsches_Eliminationsverfahren" title="Gaußsches Eliminationsverfahren">gaußschen Eliminationsverfahren</a> zum Einsatz kommen.
</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>
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<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="Ring_(Algebra)" title="Ring (Algebra)">Ring</a> 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>
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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
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</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 Standardmatrix <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_{ij}=(e_{kl})\in R^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>l</mi>
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</msub>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle E_{ij}=(e_{kl})\in R^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b416d374d37f12ae8704ca83cfc1968fdff6f8bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.307ex; height:3.009ex;" alt="{\displaystyle E_{ij}=(e_{kl})\in R^{m\times n}}" loading="lazy"></span> die <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> mit den Einträgen
</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 e_{kl}={\begin{cases}1&{\text{für }}i=k{\text{ und }}j=l\\0&{\text{sonst}}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>l</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>für </mtext>
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<mi>i</mi>
<mo>=</mo>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext> und </mtext>
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<mi>j</mi>
<mo>=</mo>
<mi>l</mi>
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<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sonst</mtext>
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</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{kl}={\begin{cases}1&{\text{für }}i=k{\text{ und }}j=l\\0&{\text{sonst}}\end{cases}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64b9eb9294ab90a6bd2ba2238e6ea8b04ea8c7ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.262ex; height:6.176ex;" alt="{\displaystyle e_{kl}={\begin{cases}1&{\text{für }}i=k{\text{ und }}j=l\\0&{\text{sonst}}\end{cases}}}" loading="lazy"></span></dd></dl>
<p>für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09a3c5f7ce97004561fe563631144135783e8164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.691ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,m}" 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 l=1,\ldots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=1,\ldots ,n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f741b515dd45d8312937a4e595f8a560cd5b8ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.527ex; height:2.509ex;" alt="{\displaystyle l=1,\ldots ,n}" loading="lazy"></span>.<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> Bei der Standardmatrix <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_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acbcb625c128efafee881204113cd9e7a8a293a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.192ex; height:2.843ex;" alt="{\displaystyle E_{ij}}" loading="lazy"></span> ist demnach der Eintrag an der Stelle <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,j)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"></span> gleich eins und alle anderen Einträge gleich null. Eine Standardmatrix wird auch als Standard-Einheitsmatrix<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> oder Matrixeinheit<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> bezeichnet und gelegentlich durch <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_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b6a2274e22dc1d2778c28f3ce5b946d90ba2756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.561ex; height:2.343ex;" alt="{\displaystyle e_{ij}}" loading="lazy"></span> statt <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_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acbcb625c128efafee881204113cd9e7a8a293a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.192ex; height:2.843ex;" alt="{\displaystyle E_{ij}}" loading="lazy"></span> notiert.
</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 Standardmatrizen 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 3\times 3}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
<mo>×<!-- × --></mo>
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" 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 E_{11}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}},E_{23}={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>11</mn>
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</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>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
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<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>12</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<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>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
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<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>23</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</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>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{11}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}},E_{23}={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18d9bc89f4543b82639bf8e096cf9f3036b4902b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:60.99ex; height:9.176ex;" alt="{\displaystyle E_{11}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}},E_{23}={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\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="Darstellungen">Darstellungen</h3></div>
<p>Jede Standardmatrix <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_{ij}\in R^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<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 E_{ij}\in R^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6efe369f9c64caf6b8e3ef18c956592f7872c9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.737ex; height:3.009ex;" alt="{\displaystyle E_{ij}\in R^{m\times n}}" loading="lazy"></span> lässt sich als <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadisches Produkt</a> der beiden 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_{i}\in R^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{i}\in R^{m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dd5c93e689cd632a3126ec37f09ce5df9d2a1a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.163ex; height:2.676ex;" alt="{\displaystyle e_{i}\in R^{m}}" 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 e_{j}\in R^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{j}\in R^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c809d9bf8075c4286a4ba2f8940c2bd04932c6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.816ex; height:3.009ex;" alt="{\displaystyle e_{j}\in R^{n}}" loading="lazy"></span> darstellen, 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 E_{ij}=e_{i}\otimes e_{j}=e_{i}\cdot (e_{j})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⊗<!-- ⊗ --></mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ij}=e_{i}\otimes e_{j}=e_{i}\cdot (e_{j})^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0802a55d878b3e567b1f93143e0563a61c38058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.86ex; height:3.343ex;" alt="{\displaystyle E_{ij}=e_{i}\otimes e_{j}=e_{i}\cdot (e_{j})^{T}}" 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^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b37958eca1d75315b998189f3b26bf163d5c52d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.473ex; height:2.676ex;" alt="{\displaystyle e^{T}}" loading="lazy"></span> der transponierte Vektor zu <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> ist. Mit Hilfe des <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Deltas</a> lässt sich eine Standardmatrix auch 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 E_{ij}=(\delta _{ik}\,\delta _{jl})_{k=1,\ldots ,m \atop l=1,\ldots ,n}=(\delta _{(i,j),(k,l)})_{k=1,\ldots ,m \atop l=1,\ldots ,n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>l</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac linethickness="0">
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>m</mi>
</mrow>
<mrow>
<mi>l</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mrow>
</mfrac>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>l</mi>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac linethickness="0">
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>m</mi>
</mrow>
<mrow>
<mi>l</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mrow>
</mfrac>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ij}=(\delta _{ik}\,\delta _{jl})_{k=1,\ldots ,m \atop l=1,\ldots ,n}=(\delta _{(i,j),(k,l)})_{k=1,\ldots ,m \atop l=1,\ldots ,n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df729244e304a60272d6f98b4c82105602b84612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:39.638ex; height:4.176ex;" alt="{\displaystyle E_{ij}=(\delta _{ik}\,\delta _{jl})_{k=1,\ldots ,m \atop l=1,\ldots ,n}=(\delta _{(i,j),(k,l)})_{k=1,\ldots ,m \atop l=1,\ldots ,n}}" loading="lazy"></span></dd></dl>
<p>notieren.
</p>
<div class="mw-heading mw-heading3"><h3 id="Symmetrie">Symmetrie</h3></div>
<p>Für die <a href="Transponierte_Matrix" title="Transponierte Matrix">Transponierte</a> einer Standardmatrix <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_{ij}\in R^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<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 E_{ij}\in R^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6efe369f9c64caf6b8e3ef18c956592f7872c9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.737ex; height:3.009ex;" alt="{\displaystyle E_{ij}\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 (E_{ij})^{T}=E_{ji}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (E_{ij})^{T}=E_{ji}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6896a775bc08d5a8089efeeb4f33b2ff6c22c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.682ex; height:3.343ex;" alt="{\displaystyle (E_{ij})^{T}=E_{ji}}" loading="lazy"></span>.</dd></dl>
<p>Damit sind nur die Standardmatrizen <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_{ii}\in R^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>i</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 E_{ii}\in R^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19e4145d3cc170856b694817735607c0761bf035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.676ex;" alt="{\displaystyle E_{ii}\in R^{n\times n}}" loading="lazy"></span> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Produkt">Produkt</h3></div>
<p>Für das <a href="Matrizenprodukt" class="mw-redirect" title="Matrizenprodukt">Produkt</a> zweier Standardmatrizen <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_{ij}\in R^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<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 E_{ij}\in R^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6efe369f9c64caf6b8e3ef18c956592f7872c9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.737ex; height:3.009ex;" alt="{\displaystyle E_{ij}\in R^{m\times n}}" 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 E_{kl}\in R^{n\times p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>l</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{kl}\in R^{n\times p}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96d7ba6aeb9a06222293f921f618747e363f46c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.223ex; height:2.676ex;" alt="{\displaystyle E_{kl}\in R^{n\times p}}" 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 E_{ij}\cdot E_{kl}={\begin{cases}E_{il}&{\text{falls }}j=k\\0&{\text{sonst,}}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>l</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>l</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls </mtext>
</mrow>
<mi>j</mi>
<mo>=</mo>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sonst,</mtext>
</mrow>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ij}\cdot E_{kl}={\begin{cases}E_{il}&{\text{falls }}j=k\\0&{\text{sonst,}}\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aab8b73a088240e6ba287539b27230e80567b00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.019ex; height:6.176ex;" alt="{\displaystyle E_{ij}\cdot E_{kl}={\begin{cases}E_{il}&{\text{falls }}j=k\\0&{\text{sonst,}}\end{cases}}}" 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 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> die <a href="Nullmatrix" title="Nullmatrix">Nullmatrix</a> 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 m\times p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\times p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e459f087ef822dc6fba54b953c60de61be69c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.05ex; height:2.009ex;" alt="{\displaystyle m\times p}" loading="lazy"></span> ist.
</p>
<div class="mw-heading mw-heading3"><h3 id="Kenngrößen"><span id="Kenngr.C3.B6.C3.9Fen"></span>Kenngrößen</h3></div>
<p>Für den <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a> einer Standardmatrix 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 {rang} (E_{ij})=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>rang</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {rang} (E_{ij})=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3d62876fc06b8ef4a6ed8252ddda67ce4fbe407.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.792ex; height:3.009ex;" alt="{\displaystyle \operatorname {rang} (E_{ij})=1}" loading="lazy"></span>.</dd></dl>
<p>Für die <a href="Determinante" title="Determinante">Determinante</a> und die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> einer quadratischen <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\times m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\times m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/367523981d714dcd9214703d654bfdedbe58d44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.921ex; height:1.676ex;" alt="{\displaystyle m\times m}" loading="lazy"></span>-Standardmatrix gilt 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 \operatorname {det} (E_{ij})={\begin{cases}1&{\text{falls }}m=1\\0&{\text{sonst}}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>det</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls </mtext>
</mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sonst</mtext>
</mrow>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (E_{ij})={\begin{cases}1&{\text{falls }}m=1\\0&{\text{sonst}}\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab9666e4482e403818767a03baa51a90b5c97da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.276ex; height:6.176ex;" alt="{\displaystyle \operatorname {det} (E_{ij})={\begin{cases}1&{\text{falls }}m=1\\0&{\text{sonst}}\end{cases}}}" 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 \operatorname {spur} (E_{ij})=\delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {spur} (E_{ij})=\delta _{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40ef7dff803f54172c9b74d7b6d761f9f7348962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.023ex; height:3.009ex;" alt="{\displaystyle \operatorname {spur} (E_{ij})=\delta _{ij}}" loading="lazy"></span>.</dd></dl>
<p>Das <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristische Polynom</a> einer quadratischen Standardmatrix <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_{ij}\in K^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ij}\in K^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93b341c8de5dbe99ed4c379143ca2a19d8a173d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.61ex; height:3.009ex;" alt="{\displaystyle E_{ij}\in K^{n\times n}}" loading="lazy"></span> über einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</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 K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> ergibt sich zu
</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 (\lambda )={\begin{cases}\lambda ^{n-1}(\lambda -1)&{\text{falls }}i=j\\\lambda ^{n}&{\text{sonst.}}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls </mtext>
</mrow>
<mi>i</mi>
<mo>=</mo>
<mi>j</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sonst.</mtext>
</mrow>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi (\lambda )={\begin{cases}\lambda ^{n-1}(\lambda -1)&{\text{falls }}i=j\\\lambda ^{n}&{\text{sonst.}}\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdd88d1ec4e5ecaf5de085647e013deb818a0236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.901ex; height:6.176ex;" alt="{\displaystyle \chi (\lambda )={\begin{cases}\lambda ^{n-1}(\lambda -1)&{\text{falls }}i=j\\\lambda ^{n}&{\text{sonst.}}\end{cases}}}" loading="lazy"></span></dd></dl>
<p>Im Fall <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\neq j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>≠<!-- ≠ --></mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\neq j}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d95aeb406bb427ac96806bc00c30c91d31b858be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\neq j}" loading="lazy"></span> ist demnach der einzige <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</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>. Für <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=j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>=</mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i=j}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/706e0928b2bf0f24076b0c90bb20616ff2068343.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i=j}" loading="lazy"></span> existiert zusätzlich noch der Eigenwert <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> mit einfacher Vielfachheit und zugehörigem <a href="Eigenvektor" class="mw-redirect" title="Eigenvektor">Eigenvektor</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_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwendung">Verwendung</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Matrixeinträge"><span id="Matrixeintr.C3.A4ge"></span>Matrixeinträge</h3></div>
<p>Mit Hilfe von Standardmatrizen <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_{ji}\in R^{n\times m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{ji}\in R^{n\times m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c23ddc8e5ac83a78493e6288c39d586a33aa7897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.737ex; height:3.009ex;" alt="{\displaystyle E_{ji}\in R^{n\times m}}" loading="lazy"></span> können auch einzelne Matrixeinträge als Spur dargestellt werden. 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 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>, dann 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 (A)_{ij}=\operatorname {spur} (AE_{ji})=\operatorname {spur} (E_{ji}A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>A</mi>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (A)_{ij}=\operatorname {spur} (AE_{ji})=\operatorname {spur} (E_{ji}A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab06f6776924b7f2f0b0fcda2a48cfb2e1e32d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.542ex; height:3.009ex;" alt="{\displaystyle (A)_{ij}=\operatorname {spur} (AE_{ji})=\operatorname {spur} (E_{ji}A)}" loading="lazy"></span>.</dd></dl>
<p>Für das Produkt zweier 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\in R^{m\times p}}">
<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>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in R^{m\times p}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98ab7eacbf468042ce5a31694095797889ee3d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.128ex; height:2.343ex;" alt="{\displaystyle A\in R^{m\times p}}" 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 B\in R^{p\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B\in R^{p\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ddcf022983e3e4dea11a3d304659dec426047ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.693ex; height:2.343ex;" alt="{\displaystyle B\in R^{p\times n}}" loading="lazy"></span> gilt 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 (AB)_{ij}=\operatorname {spur} (BE_{ji}A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>B</mi>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>B</mi>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (AB)_{ij}=\operatorname {spur} (BE_{ji}A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b44e187529b4a0bd21e6bb3382bb5b827e54b98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.814ex; height:3.009ex;" alt="{\displaystyle (AB)_{ij}=\operatorname {spur} (BE_{ji}A)}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Standardbasis">Standardbasis</h3></div>
<p>Die Menge der Standardmatrizen über einem gegebenen Körper <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{E_{ij}\in K^{m\times n}\mid i=1,\ldots ,m,j=1,\ldots ,n\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>m</mi>
<mo>,</mo>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{E_{ij}\in K^{m\times n}\mid i=1,\ldots ,m,j=1,\ldots ,n\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a7cc6895846b16fa8996f15276d5522e9141eec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.437ex; height:3.009ex;" alt="{\displaystyle \{E_{ij}\in K^{m\times n}\mid i=1,\ldots ,m,j=1,\ldots ,n\}}" loading="lazy"></span> bildet die <a href="Standardbasis" title="Standardbasis">Standardbasis</a> für den <a href="Vektorraum" title="Vektorraum">Vektorraum</a> der Matrizen. 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 K^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>K</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 K^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab303433d6aa889f1f94258f6afb6a04ffea388d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.617ex; height:2.343ex;" alt="{\displaystyle A\in K^{m\times n}}" loading="lazy"></span> lässt sich somit als <a href="Linearkombination" title="Linearkombination">Linearkombination</a> von Standardmatrizen 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 A=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}E_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</munderover>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}E_{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7ee0262e472daf6385d5d70384f1b2aee91b6f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.225ex; height:7.176ex;" alt="{\displaystyle A=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}E_{ij}}" loading="lazy"></span></dd></dl>
<p>mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}\in K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{ij}\in K}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f96a8523bede0f971a412a30e9329258b366e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.614ex; height:2.843ex;" alt="{\displaystyle a_{ij}\in K}" loading="lazy"></span> darstellen. So bilden die vier Standardmatrizen <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_{11}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>11</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{11}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dcf510f6a311251752a33ff415a05eada90ca7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.591ex; height:2.509ex;" alt="{\displaystyle E_{11}}" 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 E_{12}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>12</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{12}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb3cd3f6da6648dfca7acb94b6a48617c93fd6b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.591ex; height:2.509ex;" alt="{\displaystyle E_{12}}" 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 E_{21}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>21</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{21}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/780267746007165bb37cf5785f6fa309cdecb539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.591ex; height:2.509ex;" alt="{\displaystyle E_{21}}" 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 E_{22}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>22</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{22}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9635619bbd7a831775c5e00bf38151994264d6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.591ex; height:2.509ex;" alt="{\displaystyle E_{22}}" loading="lazy"></span> die Standardbasis des Raums der <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 (2\times 2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>×<!-- × --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (2\times 2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd967d734835dc2bf4d3f1b10707f0052a78a650.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle (2\times 2)}" loading="lazy"></span>-Matrizen und man erhält beispielsweise
</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 {\begin{pmatrix}2&4\\3&1\end{pmatrix}}={\begin{pmatrix}2&0\\0&0\end{pmatrix}}+{\begin{pmatrix}0&4\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\3&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&1\end{pmatrix}}=2E_{11}+4E_{12}+3E_{21}+1E_{22}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<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>
<mn>2</mn>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>11</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>4</mn>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>12</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>3</mn>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>21</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>1</mn>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>22</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}2&4\\3&1\end{pmatrix}}={\begin{pmatrix}2&0\\0&0\end{pmatrix}}+{\begin{pmatrix}0&4\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\3&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&1\end{pmatrix}}=2E_{11}+4E_{12}+3E_{21}+1E_{22}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bf1b995bb21854f24f85b0d2b3cd9a01512e93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:86.356ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}2&4\\3&1\end{pmatrix}}={\begin{pmatrix}2&0\\0&0\end{pmatrix}}+{\begin{pmatrix}0&4\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\3&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&1\end{pmatrix}}=2E_{11}+4E_{12}+3E_{21}+1E_{22}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Elementarmatrizen">Elementarmatrizen</h3></div>
<p>Standardmatrizen werden auch zur Darstellung der drei Typen von <a href="Elementarmatrix" title="Elementarmatrix">Elementarmatrizen</a> der Form
</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 {\begin{aligned}R_{ij}(\alpha )&=I+\alpha E_{ij}\\S_{i}(\gamma )&=I+(\gamma -1)E_{ii}\\T_{i,j}&=I-E_{ii}-E_{jj}+E_{ij}+E_{ji}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{ij}(\alpha )&=I+\alpha E_{ij}\\S_{i}(\gamma )&=I+(\gamma -1)E_{ii}\\T_{i,j}&=I-E_{ii}-E_{jj}+E_{ij}+E_{ji}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/019801f6f5f0ba87ccda901f0dd25549b2ac0cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:35.691ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}R_{ij}(\alpha )&=I+\alpha E_{ij}\\S_{i}(\gamma )&=I+(\gamma -1)E_{ii}\\T_{i,j}&=I-E_{ii}-E_{jj}+E_{ij}+E_{ji}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> als der <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> 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 \alpha ,\gamma \in K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>,</mo>
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<mi>K</mi>
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<annotation encoding="application/x-tex">{\displaystyle \alpha ,\gamma \in K}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a044e591261f9554bc218f27c52a5587511226af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.69ex; height:2.676ex;" alt="{\displaystyle \alpha ,\gamma \in K}" loading="lazy"></span> verwendet. Durch Multiplikation von links mit einer solchen Elementarmatrix werden Reihenoperationen, Skalierungen und Transpositionen an einer gegebenen Matrix durchgeführt. Diese Elementarmatrizen kommen bei der Beschreibung des <a href="Gau%C3%9Fsches_Eliminationsverfahren" title="Gaußsches Eliminationsverfahren">gaußschen Eliminationsverfahrens</a> zur Lösung <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">linearer Gleichungssysteme</a> zum Einsatz.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Tilo Arens, Frank Hettlich, Christian Karpfinger, Ulrich Kockelkorn, Klaus Lichtenegger, <a href="Hellmuth_Stachel" title="Hellmuth Stachel">Hellmuth Stachel</a>: <cite style="font-style:italic">Mathematik</cite>. 2. Auflage. Spektrum Akademischer Verlag, 2011, ISBN 3-8274-2347-3.<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:Standardmatrix&rft.au=Tilo+Arens%2C+Frank+Hettlich%2C+Christian+Karpfinger%2C+...&rft.btitle=Mathematik&rft.date=2011&rft.edition=2.&rft.genre=book&rft.isbn=3827423473&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></li>
<li><a href="Michael_Artin" title="Michael Artin">Michael Artin</a>: <cite style="font-style:italic">Algebra</cite>. Springer, 1998, ISBN 3-7643-5938-2.<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:Standardmatrix&rft.au=Michael+Artin&rft.btitle=Algebra&rft.date=1998&rft.genre=book&rft.isbn=3764359382&rft.pub=Springer" style="display:none"> </span></li>
<li>Christian Voigt, Jürgen Adamy: <cite style="font-style:italic">Formelsammlung der Matrizenrechnung</cite>. Oldenbourg, 2007, ISBN 3-486-58350-6.<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:Standardmatrix&rft.au=Christian+Voigt%2C+J%C3%BCrgen+Adamy&rft.btitle=Formelsammlung+der+Matrizenrechnung&rft.date=2007&rft.genre=book&rft.isbn=3486583506&rft.pub=Oldenbourg" style="display:none"> </span></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">Voigt, Adamy: <cite style="font-style:italic">Formelsammlung der Matrizenrechnung</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>8</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:Standardmatrix&rft.au=Voigt%2C+Adamy&rft.btitle=Formelsammlung+der+Matrizenrechnung&rft.genre=book&rft.pages=8" 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">Arens et al: <cite style="font-style:italic">Mathematik</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>508</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:Standardmatrix&rft.au=Arens+et+al&rft.btitle=Mathematik&rft.genre=book&rft.pages=508" style="display:none"> </span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Artin: <cite style="font-style:italic">Algebra</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>11</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:Standardmatrix&rft.au=Artin&rft.btitle=Algebra&rft.genre=book&rft.pages=11" style="display:none"> </span></span>
</li>
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