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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Matrixpotenz</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>In der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> bezeichnet die <b>Matrixpotenz</b> das Ergebnis einer wiederholten <a href="Matrixmultiplikation" class="mw-redirect" title="Matrixmultiplikation">Matrixmultiplikation</a>.

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

<p>Die Potenz einer <a href="Quadratische_Matrix" class="mw-redirect" title="Quadratische Matrix">quadratischen 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 R^{n\times n}}">

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

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</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> über einem <a href="Halbring_(Algebraische_Struktur)" class="mw-redirect" title="Halbring (Algebraische Struktur)">Halbring</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}">

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

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</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> wird analog zur <a href="Potenz_(Mathematik)" title="Potenz (Mathematik)">Potenz</a> einer Zahl als wiederholte Multiplikation definiert. 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}">

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

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

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<annotation encoding="application/x-tex">{\displaystyle A^{0}=E;\quad A^{1}=A;\quad A^{2}=A\cdot A;\quad A^{3}=A\cdot A\cdot A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a530e7bd5193535638825c9671073b33e4770a21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:50.924ex; height:3.009ex;" alt="{\displaystyle A^{0}=E;\quad A^{1}=A;\quad A^{2}=A\cdot A;\quad A^{3}=A\cdot A\cdot A}" loading="lazy"></span> usw.</dd></dl>

<p>Allgemein:

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<annotation encoding="application/x-tex">{\displaystyle A^{n}=\underbrace {A\cdot A\dotsc A} _{n{\text{-mal}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c41940a6904964c4d9933daaf7ac356a9885b0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-right: -0.028ex; width:16.494ex; height:5.676ex;" alt="{\displaystyle A^{n}=\underbrace {A\cdot A\dotsc A} _{n{\text{-mal}}}}" loading="lazy"></span>.</dd></dl>

<p>Formal definiert man die Potenz <a href="Rekursive_Definition" class="mw-redirect" title="Rekursive Definition">rekursiv</a>:

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}">

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

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

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eca37929de83d16c86cda063da3ff6544c582fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.671ex; height:2.676ex;" alt="{\displaystyle A^{0}=E}" loading="lazy"></span> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> und</li>

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<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} =\{0,1,2,\dotsc \}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09ae813090627f7868e57e8aed87cc9adefc1396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.466ex; height:2.843ex;" alt="{\displaystyle k\in \mathbb {N} =\{0,1,2,\dotsc \}}" loading="lazy"></span> gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{k+1}=A^{k}\cdot A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cb2769e4f88efb35017c578c668ee4d8aa5b685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.285ex; height:2.676ex;" alt="{\displaystyle A^{k+1}=A^{k}\cdot A}" loading="lazy"></span>.</li></ul>

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

<p>Es gelten die <a href="Potenz_(Mathematik)#Potenzgesetze" title="Potenz (Mathematik)">Potenzgesetze</a>:

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 n,m\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n,m\in \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a011b8508a4abb5730a00fa6c158c79248c34f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.988ex; height:2.509ex;" alt="{\displaystyle n,m\in \mathbb {N} }" loading="lazy"></span> gilt

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<annotation encoding="application/x-tex">{\displaystyle A^{n+m}=A^{n}\cdot A^{m}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf1d3d9fce09cf11b9b391cd125fd2bc4a240ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.84ex; height:2.509ex;" alt="{\displaystyle A^{n+m}=A^{n}\cdot A^{m}}" loading="lazy"></span>,</li>

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<annotation encoding="application/x-tex">{\displaystyle A^{n\cdot m}=\left(A^{n}\right)^{m}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94367eac81d78cb526851df23e81566d1063b053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.406ex; height:3.009ex;" alt="{\displaystyle A^{n\cdot m}=\left(A^{n}\right)^{m}}" loading="lazy"></span>.</li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="Negative_Exponenten">Negative Exponenten</h3></div>

<p>Für <a href="Regul%C3%A4re_Matrix" title="Reguläre Matrix">invertierbare Matrizen</a> sind auch Potenzen mit negativen ganzzahligen Exponenten definiert. Die Schreibweise <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}}">

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

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</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> für die <a href="Inverse_Matrix" title="Inverse Matrix">inverse Matrix</a> kann auch als Matrixpotenz interpretiert werden. Für negative Exponenten <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f00139753ecf4fe00a10a17bd5620b70a61b29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"></span>, setzt man

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<annotation encoding="application/x-tex">{\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7d3b01ba9ec1ac82badfdc7b3162aea150897cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.763ex; height:3.343ex;" alt="{\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Gebrochene_Exponenten">Gebrochene Exponenten</h3></div>

<p>Matrixpotenzen mit nicht ganzzahligen Exponenten, beispielsweise die <a href="Quadratwurzel_einer_Matrix" title="Quadratwurzel einer Matrix">Quadratwurzel einer Matrix</a>, können nur in Sonderfällen definiert werden.

</p><p>In manchen Fällen kann die Matrixpotenz auf die Potenz von <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> zurückgeführt werden.

Lässt sich die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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

existieren also eine reguläre 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 T}">

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

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</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> und eine Diagonalmatrix <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 D}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> mit

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<annotation encoding="application/x-tex">{\displaystyle A=T\cdot D\cdot T^{-1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1183ba4192dd415f611f822c0043be867bb81b6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.813ex; height:2.676ex;" alt="{\displaystyle A=T\cdot D\cdot T^{-1}}" loading="lazy"></span>

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

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> ist <a href="%C3%84hnlichkeit_(Matrix)" title="Ähnlichkeit (Matrix)">ähnlich</a> 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 D}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>),

so gilt

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<annotation encoding="application/x-tex">{\displaystyle A^{n}=T\cdot D^{n}\cdot T^{-1}\ .}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b42aa23d38ea29fbc70ea7a95de9a1d1b71b304.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.478ex; height:2.676ex;" alt="{\displaystyle A^{n}=T\cdot D^{n}\cdot T^{-1}\ .}" loading="lazy"></span></dd></dl>

<p>Die Potenz einer Diagonalmatrix erhält man durch Potenzieren der Diagonalelemente.

Sind die Diagonalelemente von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> (also die <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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

so bleiben obige Potenzgesetze auch für gebrochene Exponenten gültig.

</p><p>Wenn sich eine Matrix nicht diagonalisieren lässt,

so findet man eine sinnvolle Verallgemeinerung der Matrixpotenz über die <a href="Binomische_Reihe" title="Binomische Reihe">binomische Reihe</a>.

Eine schnelle Berechnungsmethode für diese Verallgemeinerung erhält man über die <a href="Jordansche_Normalform" title="Jordansche Normalform">Jordansche Normalform</a>.

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=T\cdot J\cdot T^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle A=T\cdot J\cdot T^{-1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a948b25261b7f12b6093b294ce73fd915462a04c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.36ex; height:2.676ex;" alt="{\displaystyle A=T\cdot J\cdot T^{-1}}" loading="lazy"></span> eine Jordanzerlegung,

dann gilt

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<annotation encoding="application/x-tex">{\displaystyle A^{n}=T\cdot J^{n}\cdot T^{-1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d89de0a2fb32ebdd640feb067240a948ae59893.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.852ex; height:2.676ex;" alt="{\displaystyle A^{n}=T\cdot J^{n}\cdot T^{-1}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Effiziente_Berechnung">Effiziente Berechnung</h2></div>

<p>Ist der Exponent eine <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahl</a>,

so lässt sich die Matrixpotenz effizient mit <a href="Bin%C3%A4re_Exponentiation" title="Binäre Exponentiation">binärer Exponentiation</a> berechnen.

Die Einschränkungen an den Zahlenbereich der Matrixelemente sind gering:

</p>

<ul><li>Ist der Exponent nicht-negativ, so müssen die Matrixelemente in einem Ring liegen.</li>

<li>Ist der Exponent negativ, so müssen die Matrixelemente in einem Körper liegen.</li></ul>

<p>Ist der Zahlenbereich der Matrixelemente <a href="Algebraischer_Abschluss" title="Algebraischer Abschluss">algebraisch abgeschlossen</a>,

kann man also darin beliebige <a href="Algebraische_Gleichung" title="Algebraische Gleichung">algebraische Gleichungen</a> lösen,

so kann der Exponent auch rational sein

und die Matrixpotenz kann über die <a href="Jordansche_Normalform" title="Jordansche Normalform">Jordansche Normalform</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</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>

auf Potenzen von skalaren Werten zurückgeführt werden, siehe <a href="#Gebrochene_Exponenten">oben</a>.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Polynome_und_Potenzreihen">Polynome und Potenzreihen</h3></div>

<p>Mittels der Matrixpotenz lassen sich <a href="Polynom" title="Polynom">Polynome</a> auch für Matrizen definieren. Ein Beispiel dafür ist z. B. das <a href="Minimalpolynom_(Lineare_Algebra)" title="Minimalpolynom (Lineare Algebra)">Minimalpolynom</a>.

Genauso kann man auch <a href="Potenzreihe" title="Potenzreihe">Potenzreihen</a> für Matrizen definieren, die wichtigsten Reihen sind dabei der <a href="Matrixlogarithmus" title="Matrixlogarithmus">Matrixlogarithmus</a>, das <a href="Matrixexponential" title="Matrixexponential">Matrixexponential</a> sowie die <a href="Neumann-Reihe" title="Neumann-Reihe">Neumann-Reihe</a>.

</p>

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

<p>Durch geeignete Wahl des zugrunde liegenden <a href="Halbring_(Algebraische_Struktur)" class="mw-redirect" title="Halbring (Algebraische Struktur)">Halbrings</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}">

</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> lässt sich das Finden der <a href="K%C3%BCrzester_Pfad" title="Kürzester Pfad">kürzesten Pfade</a> in einem <a href="Graph_(Graphentheorie)" title="Graph (Graphentheorie)">Graphen</a> auf die Berechnung einer Potenz der <a href="Adjazenzmatrix" title="Adjazenzmatrix">Adjazenzmatrix</a> des Graphen zurückführen. Die <a href="Min-Plus-Matrixmultiplikations-Algorithmus" title="Min-Plus-Matrixmultiplikations-Algorithmus">Min-Plus-Matrixmultiplikation</a> erhält man, indem man als <a href="Tr%C3%A4germenge" title="Trägermenge">Trägermenge</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> die <a href="Erweiterte_reelle_Zahl" title="Erweiterte reelle Zahl">erweiterten reellen Zahlen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\ast }=\mathbb {R} ^{+}\cup \{\infty \}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\ast }=\mathbb {R} ^{+}\cup \{\infty \}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cb4d7dcbb2952c7bd7aed6989166423dea30100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.251ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{\ast }=\mathbb {R} ^{+}\cup \{\infty \}}" loading="lazy"></span> wählt. Die Addition in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> entspricht dann der <a href="Gr%C3%B6%C3%9Ftes_und_kleinstes_Element" title="Größtes und kleinstes Element">Minimumbildung</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\ast }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7bdfbbb4dfe751816139d019d2719a6447c5147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\ast }}" loading="lazy"></span> und die Multiplikation in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> der Addition in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\ast }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7bdfbbb4dfe751816139d019d2719a6447c5147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\ast }}" loading="lazy"></span>, wobei man <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+\infty =\infty +x=\infty }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1125c5ac36160cea79d380d2cd6237972d716012.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.508ex; height:2.176ex;" alt="{\displaystyle x+\infty =\infty +x=\infty }" loading="lazy"></span> setzt. Die absorbierende Null in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> ist dann <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 \infty }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"></span>, während das Einselement in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> durch die Zahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</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> dargestellt wird. Ist nun <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\in R^{n\times n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fde8b67edba12a5a2332ca81d7a07db4a9beab5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.154ex; height:2.343ex;" alt="{\displaystyle K\in R^{n\times n}}" loading="lazy"></span> die Kostenmatrix eines Graphen 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 n}">

</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> <a href="Knoten_(Graphentheorie)" title="Knoten (Graphentheorie)">Knoten</a>, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle D=\sum _{k=0}^{n}K^{k}}">

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle D=\sum _{k=0}^{n}K^{k}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07758ec711a1d3eeda762a6c4cb562aa24efd29d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.236ex; height:3.176ex;" alt="{\displaystyle \textstyle D=\sum _{k=0}^{n}K^{k}}" loading="lazy"></span> die zugehörige Entfernungsmatrix mit den Längen der kürzesten Pfade zwischen allen Knoten des Graphen. Da die Addition in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> idempotent ist, 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 D=(1+K)^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=(1+K)^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8c60712788803e7b5d747d15e9b65ee7f544c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.119ex; height:2.843ex;" alt="{\displaystyle D=(1+K)^{n}}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Weitere_Anwendungen">Weitere Anwendungen</h3></div>

<ul><li>In der theoretischen Ökonomie bzw. <a href="Biologie" title="Biologie">Biologie</a> werden Matrixpotenzen zur Analyse langfristiger <a href="Population_(Biologie)" title="Population (Biologie)">Populationsentwicklungen</a> eingesetzt, beispielsweise unter Nutzung einer <a href="Leslie-Matrix" title="Leslie-Matrix">Leslie-Matrix</a>.<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></li>

<li>Des Weiteren gibt es Anwendungen bei der <a href="Stereobasisverbreiterung" title="Stereobasisverbreiterung">Stereobasisverbreiterung</a>.</li></ul>

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

<ul><li><a href="Peter_Knabner" title="Peter Knabner">Peter Knabner</a>, <a href="Wolf_Barth_(Mathematiker)" title="Wolf Barth (Mathematiker)">Wolf Barth</a>: <cite style="font-style:italic">Lineare Algebra. Grundlagen und Anwendungen</cite> (= <cite style="font-style:italic">Springer-Lehrbuch</cite>). Springer Spektrum, Berlin u. a. 2013, ISBN 978-3-642-32185-6.<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:Matrixpotenz&rft.au=Peter+Knabner%2C+Wolf+Barth&rft.btitle=Lineare+Algebra.+Grundlagen+und+Anwendungen&rft.date=2013&rft.genre=book&rft.isbn=9783642321856&rft.place=Berlin+u.+a.&rft.pub=Springer+Spektrum&rft.series=Springer-Lehrbuch" style="display:none"> </span></li>

<li><a href="Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a>: <cite style="font-style:italic">Lineare Algebra</cite>. Eine Einführung für Studienanfänger (= <cite style="font-style:italic">Springer-Lehrbuch</cite>). Springer, Berlin 2003, ISBN 3-540-43949-8 (englisch: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Introduction to linear algebra</cite>. Übersetzt von Michael Dellnitz).<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:Matrixpotenz&rft.au=Gilbert+Strang&rft.btitle=Lineare+Algebra&rft.date=2003&rft.genre=book&rft.isbn=3540439498&rft.place=Berlin&rft.pub=Springer&rft.series=Springer-Lehrbuch" style="display:none"> </span></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180304054805/https://www.nibis.de/~lbs-gym/Vektorpdf2/Populationsentwicklung2.pdf"><i>Populationsentwicklung.</i></a> (PDF; 72 kB) Archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">

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</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=https%3A%2F%2Fwww.nibis.de%2F%7Elbs-gym%2FVektorpdf2%2FPopulationsentwicklung2.pdf">Original</a></span> am <span style="white-space:nowrap;">4. März 2018</span><span>;</span><span class="Abrufdatum"> abgerufen am 27. Februar 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AMatrixpotenz&rft.title=Populationsentwicklung&rft.description=Populationsentwicklung&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20180304054805%2Fhttps%3A%2F%2Fwww.nibis.de%2F%7Elbs-gym%2FVektorpdf2%2FPopulationsentwicklung2.pdf&rft.source=https://www.nibis.de/~lbs-gym/Vektorpdf2/Populationsentwicklung2.pdf"> </span>, Archivlink</span>

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