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<title>Matrixexponential</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Matrixexponential</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">In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist das Matrixexponential, auch als Matrixexponentialfunktion bezeichnet eine <a href="Matrixfunktion" title="Matrixfunktion">Matrixfunktion</a>, welche analog zur gewöhnlichen (skalaren) <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> definiert ist. Das Matrixexponential stellt die Verbindung zwischen <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a> und der zugehörigen <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a> her.

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

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> eine <a href="Reelle_Zahl" title="Reelle Zahl">reelle</a> oder <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-<a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>. Das Exponential von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, welches mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6400aec0e00b4d4a468a64d93a97b24a9ab25a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.716ex; height:2.676ex;" alt="{\displaystyle e^{X}}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(X)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91cd111ee9dcf82fd9d42b526c7cccc5066bff12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.342ex; height:2.843ex;" alt="{\displaystyle \exp(X)}" loading="lazy"> bezeichnet wird, ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrix, welche durch die folgende <a href="Potenzreihe" title="Potenzreihe">Potenzreihe</a> definiert ist (<a href="Taylorreihe" title="Taylorreihe">Taylor-Entwicklung</a>):

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<annotation encoding="application/x-tex">{\displaystyle e^{X}=\sum _{k=0}^{\infty }{\frac {X^{k}}{k!}}=E+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\dots =E+X+{\frac {X^{2}}{2}}+{\frac {X^{3}}{6}}+\dots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0becad0faea16ecab14346702496b56f9cc0c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:70.905ex; height:7.009ex;" alt="{\displaystyle e^{X}=\sum _{k=0}^{\infty }{\frac {X^{k}}{k!}}=E+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\dots =E+X+{\frac {X^{2}}{2}}+{\frac {X^{3}}{6}}+\dots }" loading="lazy">.</dd></dl>

Diese Reihe konvergiert, genauso wie die der <a href="Exponentialfunktion" title="Exponentialfunktion">gewöhnlichen Exponentialfunktion</a>, immer. Daher ist das Exponential von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> <a href="Wohldefiniertheit" title="Wohldefiniertheit">wohldefiniert</a>. Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b4bf91a527dc01af9ef6ace81199becf1308e00.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 1\times 1}" loading="lazy">-Matrix ist, entspricht das Matrixexponential von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> der gewöhnlichen <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a>. Eine Verallgemeinerung, welche auch für unendliche Matrizen sinnvoll ist, ist die <a href="Exponentialfunktion#Exponentialfunktion_auf_beliebigen_Banachalgebren" title="Exponentialfunktion">Exponentialfunktion auf beliebigen Banachalgebren</a>.

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

Das Matrixexponential teilt eine Reihe der Eigenschaften der gewöhnlichen Exponentialfunktion. Beispielsweise ist das Exponential der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\times n)}">

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

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</math><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"> gleich der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\times n)}">

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92e07e4714afff329f71fb8accae87bc46f6a046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.012ex; height:2.676ex;" alt="{\displaystyle e^{0}=E}" loading="lazy">.</dd></dl>

Für beliebige komplexe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und beliebige komplexe Zahlen <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> gilt

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<annotation encoding="application/x-tex">{\displaystyle e^{aX}\cdot e^{bX}=e^{(a+b)X}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc08e92a075ef54ef01ddf7df7015f7bba8dbedb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.633ex; height:2.843ex;" alt="{\displaystyle e^{aX}\cdot e^{bX}=e^{(a+b)X}}" loading="lazy">.</dd></dl>

Daraus folgt

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<annotation encoding="application/x-tex">{\displaystyle e^{X}\cdot e^{-X}=e^{(1-1)X}=e^{0}=E}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11e26a42652b954877def0f8b3e7e6957e1ae48f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:28.516ex; height:2.843ex;" alt="{\displaystyle e^{X}\cdot e^{-X}=e^{(1-1)X}=e^{0}=E}" loading="lazy">,</dd></dl>

das heißt

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/558f4ff7a4df069a49f508cfba3cea043551611f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.271ex; height:3.843ex;" alt="{\displaystyle \left(e^{X}\right)^{-1}=e^{-X}}" loading="lazy">.</dd></dl>

Dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(e^{X}\right)^{-1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84edf2b473c02225bd890944c880cc5ce16a6969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.178ex; height:3.843ex;" alt="{\displaystyle \left(e^{X}\right)^{-1}}" loading="lazy"> die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6400aec0e00b4d4a468a64d93a97b24a9ab25a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.716ex; height:2.676ex;" alt="{\displaystyle e^{X}}" loading="lazy"> <a href="Inverse_Matrix" title="Inverse Matrix">inverse Matrix</a>.

Die Exponentialfunktion erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x+y}=e^{x}\,e^{y}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{x+y}=e^{x}\,e^{y}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f1b0e8d7a89c6b8a11cdabe67bf0c2af7cafd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.226ex; height:2.509ex;" alt="{\displaystyle e^{x+y}=e^{x}\,e^{y}}" loading="lazy"> für alle Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Dasselbe gilt für kommutierende Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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<annotation encoding="application/x-tex">{\displaystyle X\cdot Y=Y\cdot X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/698db8005169cdc787065271fa9a3f45e75d7193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.963ex; height:2.176ex;" alt="{\displaystyle X\cdot Y=Y\cdot X}" loading="lazy"></dd></dl>

folgt

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<annotation encoding="application/x-tex">{\displaystyle e^{X+Y}=e^{X}\cdot e^{Y}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4416374b69197fdb907253d748b41f44cd2eb0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.311ex; height:2.676ex;" alt="{\displaystyle e^{X+Y}=e^{X}\cdot e^{Y}}" loading="lazy">.</dd></dl>

Für nichtkommutierende Matrizen stimmt diese Gleichung im Allgemeinen nicht. In diesem Fall kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X+Y}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{X+Y}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acc4f22a90b0b91696beba01b7e5833f32d96ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.248ex; height:2.676ex;" alt="{\displaystyle e^{X+Y}}" loading="lazy"> mit Hilfe der <a href="Baker-Campbell-Hausdorff-Formel" title="Baker-Campbell-Hausdorff-Formel">Baker-Campbell-Hausdorff-Formel</a> berechnen.

Das Exponential der zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> <a href="Transponierte_Matrix" title="Transponierte Matrix">transponierten Matrix</a> ist gleich der Transposition des Exponentials von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(X^{\mathrm {T} }\right)=\left(\exp X\right)^{\mathrm {T} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33440a011e4a18577a7afa3670f1796f893ce259.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.344ex; height:3.509ex;" alt="{\displaystyle \exp \left(X^{\mathrm {T} }\right)=\left(\exp X\right)^{\mathrm {T} }}" loading="lazy"></dd></dl>

Daraus folgt, dass die Matrixexponentialfunktion <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische Matrizen</a> auf symmetrische Matrizen und <a href="Schiefsymmetrische_Matrix" title="Schiefsymmetrische Matrix">schiefsymmetrische Matrizen</a> auf <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrizen</a> abbildet. Analog gilt zwischen <a href="Adjungierte_Matrix" title="Adjungierte Matrix">Adjunktion</a> und Exponentiation die Beziehung

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(X^{*}\right)=\left(\exp X\right)^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/443f6ce7a110f45472102a3f0f191598d96f8f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.294ex; height:3.009ex;" alt="{\displaystyle \exp \left(X^{*}\right)=\left(\exp X\right)^{*}}" loading="lazy">,</dd></dl>

so dass die Matrixexponentialfunktion <a href="Hermitesche_Matrix" title="Hermitesche Matrix">hermitesche Matrizen</a> auf hermitesche Matrizen und <a href="Schiefhermitesche_Matrix" title="Schiefhermitesche Matrix">schiefhermitesche Matrizen</a> auf <a href="Unit%C3%A4re_Matrix" title="Unitäre Matrix">unitäre Matrizen</a> abbildet.

Weiterhin gelten:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> <a href="Invertierbare_Matrix" class="mw-redirect" title="Invertierbare Matrix">invertierbar</a> ist, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{YXY^{-1}}=Ye^{X}Y^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{YXY^{-1}}=Ye^{X}Y^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30a1263f1d4e742a629aa73aeb9e751f4c48433e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.004ex; height:3.009ex;" alt="{\displaystyle e^{YXY^{-1}}=Ye^{X}Y^{-1}}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle \det(e^{X})=e^{\operatorname {tr} (X)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f46b2b414f4bb5da941df1e777da9ba697b46e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.133ex; height:3.343ex;" alt="{\displaystyle \det(e^{X})=e^{\operatorname {tr} (X)}}" loading="lazy">, hier bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (X)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ec9387059621ca8c7a141834728fd18308f0010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.606ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (X)}" loading="lazy"> die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> der quadratischen Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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<annotation encoding="application/x-tex">{\displaystyle e^{\operatorname {diag} (x_{1},\ldots ,x_{n})}=\operatorname {diag} \left(e^{x_{1}},\ldots ,e^{x_{n}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c3fe38e51afe0ca78e769e488982c3dc2afe902.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.786ex; height:3.343ex;" alt="{\displaystyle e^{\operatorname {diag} (x_{1},\ldots ,x_{n})}=\operatorname {diag} \left(e^{x_{1}},\ldots ,e^{x_{n}}\right)}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Die_Exponentialabbildung">Die Exponentialabbildung</h2></div>

Das Exponential einer Matrix ist immer eine <a href="Invertierbare_Matrix" class="mw-redirect" title="Invertierbare Matrix">invertierbare Matrix</a>. Die <a href="Inverse_Matrix" title="Inverse Matrix">Inverse</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6400aec0e00b4d4a468a64d93a97b24a9ab25a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.716ex; height:2.676ex;" alt="{\displaystyle e^{X}}" loading="lazy"> ist durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-X}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/245fac613ff42e1476625f60938f2308645f2182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.994ex; height:2.676ex;" alt="{\displaystyle e^{-X}}" loading="lazy"> gegeben. Das (komplexe) Matrixexponential liefert somit eine Abbildung

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<annotation encoding="application/x-tex">{\displaystyle \exp \colon M_{n}(\mathbb {C} )\to {\mbox{GL}}(n,\mathbb {C} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0ec994e7437c7b5a857927ccc2e7ffc9457d1e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.354ex; height:2.843ex;" alt="{\displaystyle \exp \colon M_{n}(\mathbb {C} )\to {\mbox{GL}}(n,\mathbb {C} )}" loading="lazy"></dd></dl>

aus dem <a href="Vektorraum" title="Vektorraum">Vektorraum</a> aller (komplexen) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\times n)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0e369b52ee16c33d83f7cd0eb0f562fd91b7f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.439ex; height:2.843ex;" alt="{\displaystyle (n\times n)}" loading="lazy">-Matrizen in die <a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeine lineare Gruppe</a>, die <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> aller (komplexen) invertierbaren Matrizen. Diese Abbildung ist kein <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismus</a> auf der gesamten Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{n}(\mathbb {C} )}">

<msub>

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</msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91043943830e759fc4d53f58204701c8b3ff4d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.96ex; height:2.843ex;" alt="{\displaystyle M_{n}(\mathbb {C} )}" loading="lazy">, aber auf jeder Untergruppe, deren Matrizen multiplikativ miteinander kommutieren.

Des Weiteren ist sie <a href="Surjektiv" class="mw-redirect" title="Surjektiv">surjektiv</a>, das heißt, jede (reelle oder komplexe) invertierbare Matrix kann als die Exponentialmatrix einer komplexen Matrix geschrieben werden. <a href="Urbild_(Mathematik)" title="Urbild (Mathematik)">Urbilder</a> (bzw. lokale <a href="Koretraktion" class="mw-redirect" title="Koretraktion">Schnitte</a>) lassen sich durch <a href="Matrixlogarithmus" title="Matrixlogarithmus">Matrixlogarithmen</a> berechnen.

Für je zwei Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \|e^{X+Y}-e^{X}\|\leq \|Y\|e^{\|X\|}e^{\|Y\|}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db4e0655c61b0c94c1e9061c17a82687cae2b533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.899ex; height:3.343ex;" alt="{\displaystyle \|e^{X+Y}-e^{X}\|\leq \|Y\|e^{\|X\|}e^{\|Y\|}}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}" loading="lazy"> eine beliebige <a href="Matrixnorm" title="Matrixnorm">Matrixnorm</a> bezeichnet. Daraus folgt, dass die Exponentialabbildung <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a> und auf <a href="Kompakter_Raum" title="Kompakter Raum">kompakten</a> Teilmengen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{n}(\mathbb {C} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91043943830e759fc4d53f58204701c8b3ff4d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.96ex; height:2.843ex;" alt="{\displaystyle M_{n}(\mathbb {C} )}" loading="lazy"> sogar <a href="Lipschitz-Stetigkeit" class="mw-redirect" title="Lipschitz-Stetigkeit">lipschitzstetig</a> ist.

Für die Norm des Matrixexponentials selbst gibt es aber eine präzisere Schranke

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<annotation encoding="application/x-tex">{\displaystyle \|e^{X}\|\leq e^{\mu (X)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d8e2c8a9a2b8ee5c2ed38e059bf42f820cfe6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.126ex; height:3.343ex;" alt="{\displaystyle \|e^{X}\|\leq e^{\mu (X)}}" loading="lazy"></dd></dl>

mit der logarithmischen Matrixnorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> und dem <a href="Numerischer_Wertebereich" title="Numerischer Wertebereich">numerischen Wertebereich</a>.

Die Zuordnung

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<annotation encoding="application/x-tex">{\displaystyle t\mapsto e^{tX},\qquad t\in \mathbb {R} ,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b06b802ccb0b0b02c8ff238e92e05ea2eb9ea62c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.448ex; height:3.009ex;" alt="{\displaystyle t\mapsto e^{tX},\qquad t\in \mathbb {R} ,}" loading="lazy"></dd></dl>

definiert eine glatte <a href="Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurve</a> in der allgemeinen linearen Gruppe, welche für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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

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Dies liefert eine <a href="Einparameter-Untergruppe" title="Einparameter-Untergruppe">Einparameter-Untergruppe</a> der allgemeinen linearen Gruppe, da

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<annotation encoding="application/x-tex">{\displaystyle e^{tX}e^{sX}=e^{(t+s)X}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fb54988002481f52e233e8fc413ebb8f0813293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.533ex; height:2.843ex;" alt="{\displaystyle e^{tX}e^{sX}=e^{(t+s)X}}" loading="lazy"></dd></dl>

gilt. Die Ableitung dieser Funktion im Punkt <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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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}e^{tX}=Xe^{tX}\qquad (1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0e79c69d8e432cd4fc7e2aa2978c9f936a8a232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.206ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}e^{tX}=Xe^{tX}\qquad (1)}" loading="lazy"></dd></dl>

gegeben. Die Ableitung für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> ist gerade die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, das heißt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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Allgemeiner gilt:

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<msup>

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</msup>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}e^{X(t)}=\int _{0}^{1}e^{(1-\alpha )X(t)}{\frac {dX(t)}{dt}}e^{\alpha X(t)}d\alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70f1fd71a29c3b2bb1d60a90c04b852ca975134b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.53ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dt}}e^{X(t)}=\int _{0}^{1}e^{(1-\alpha )X(t)}{\frac {dX(t)}{dt}}e^{\alpha X(t)}d\alpha }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiele_von_Lie-Algebren_und_zugehörigen_Lie-Gruppen">Beispiele von Lie-Algebren und zugehörigen Lie-Gruppen</h3></div>

<th><a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a></th>

<th>Beispiel

</th></tr>

<tr>

<td><a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">Allgemeine lineare Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (n,K)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n,K)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.581ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n,K)}" loading="lazy">

</td>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}(n)=M_{n}(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/898f23f2161a976d2b0b07bb659179f6e68b50c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.086ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {gl}}(n)=M_{n}(\mathbb {R} )}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n)=\{X\in M_{n}(\mathbb {R} )\vert \det(X)\neq 0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c66cf01d25d6aca16c879c0cd47ffe526a7a7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.999ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n)=\{X\in M_{n}(\mathbb {R} )\vert \det(X)\neq 0\}}" loading="lazy">.

</td></tr>

<tr>

<td><a href="Orthogonale_Gruppe" title="Orthogonale Gruppe">Orthogonale Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (n,K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (n,K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc16d0ff893530ff3e3cb30d6a35686f6b8eb072.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.112ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (n,K)}" loading="lazy">

</td>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {o}}(n)=\{X\in M_{n}(\mathbb {R} )\vert X^{T}=-X\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd3b351bbcffa58910be39cb1022815f3f55ff8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.464ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {o}}(n)=\{X\in M_{n}(\mathbb {R} )\vert X^{T}=-X\}}" loading="lazy">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (n)=\{X\in M_{n}(\mathbb {R} )\vert X^{T}=X^{-1}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b6339311e9e67bce47ce6d403b1f57d47eef7f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.677ex; height:3.176ex;" alt="{\displaystyle \mathrm {O} (n)=\{X\in M_{n}(\mathbb {R} )\vert X^{T}=X^{-1}\}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Unit%C3%A4re_Gruppe" title="Unitäre Gruppe">Unitäre Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {U} (n)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a32fa84df5de5dfa91b6bdc88fb03fc8792c9f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \mathrm {U} (n)}" loading="lazy">

</td>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {u}}(n)=\{X\in M_{n}(\mathbb {C} )\vert X^{*}=-X\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a1b585d49f5ce227d4253470d6539e6b0c10da3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.194ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {u}}(n)=\{X\in M_{n}(\mathbb {C} )\vert X^{*}=-X\}}" loading="lazy">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {U} (n)=\{X\in M_{n}(\mathbb {C} )\vert X^{*}=X^{-1}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b8f9c4fffc604c44ddc33ebe19a425c534ddc0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.277ex; height:3.176ex;" alt="{\displaystyle \mathrm {U} (n)=\{X\in M_{n}(\mathbb {C} )\vert X^{*}=X^{-1}\}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Spezielle_unit%C3%A4re_Gruppe" title="Spezielle unitäre Gruppe">Spezielle unitäre Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (n)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8a205091aabd5690efdfeb7354a55844f2eb31b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (n)}" loading="lazy">

</td>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)=\{X\in M_{2}(\mathbb {C} )\vert X^{*}=-X,\operatorname {tr} (X)=0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8261b27300092d74a848626c8b58a49b794563c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:43.769ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)=\{X\in M_{2}(\mathbb {C} )\vert X^{*}=-X,\operatorname {tr} (X)=0\}}" loading="lazy"> wird von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d1185b570f67b4221307626254f64f9e619e769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }" loading="lazy"> surjektiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (2)=\{A\in M_{2}(\mathbb {C} )\vert A^{*}=A^{-1},\det(A)=1\}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)=\{A\in M_{2}(\mathbb {C} )\vert A^{*}=A^{-1},\det(A)=1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/463cf04380d11d03f21d7c152f009bb5504f86b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.506ex; height:3.176ex;" alt="{\displaystyle \mathrm {SU} (2)=\{A\in M_{2}(\mathbb {C} )\vert A^{*}=A^{-1},\det(A)=1\}}" loading="lazy"> abgebildet.

</td></tr>

<tr>

<td><a href="Spezielle_orthogonale_Gruppe" class="mw-redirect" title="Spezielle orthogonale Gruppe">Spezielle orthogonale Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n,K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n,K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddad60edb811092c81237e24d5cc1b11b4b97d6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.405ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n,K)}" loading="lazy">

</td>

</mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(n,\mathbb {R} )={\mathfrak {o}}(n,\mathbb {R} )=\{X\in M_{n}(\mathbb {R} )\vert X^{T}=-X\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d728fb87160a2c5091f5ff3cf1ac4710fd58b7b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:46.399ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {so}}(n,\mathbb {R} )={\mathfrak {o}}(n,\mathbb {R} )=\{X\in M_{n}(\mathbb {R} )\vert X^{T}=-X\}}" loading="lazy"> (<a href="Schiefsymmetrische_Matrix" title="Schiefsymmetrische Matrix">schiefsymmetrische Matrizen</a>) wird von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d1185b570f67b4221307626254f64f9e619e769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }" loading="lazy"> surjektiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n,\mathbb {R} )=\{A\in M_{n}(\mathbb {R} )\vert A^{T}=A^{-1},\det(A)=1\}}">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n,\mathbb {R} )=\{A\in M_{n}(\mathbb {R} )\vert A^{T}=A^{-1},\det(A)=1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/716abd8acb0f9548f286af08fb33e5c8fdfaa282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.014ex; height:3.176ex;" alt="{\displaystyle \mathrm {SO} (n,\mathbb {R} )=\{A\in M_{n}(\mathbb {R} )\vert A^{T}=A^{-1},\det(A)=1\}}" loading="lazy"> abgebildet.

</td></tr>

<tr>

<td><a href="Spezielle_lineare_Gruppe" title="Spezielle lineare Gruppe">Spezielle lineare Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (n,K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (n,K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dd77477c38e36ab0e5e1ea4453ce8ccc040be73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.049ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (n,K)}" loading="lazy">

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80425b262aec9e7559040696a7443754c4bb3b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:7.406ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}" loading="lazy"> wird von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d1185b570f67b4221307626254f64f9e619e769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }" loading="lazy"> nicht surjektiv auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (2,\mathbb {C} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbb {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/000a9ac5731338fd38b14f08bfaf6814f2aec65f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbb {C} )}" loading="lazy"> abgebildet. Notorisches Gegenbeispiel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}-1&a\\0&-1\end{pmatrix}}\in \mathrm {SL} (2,\mathbb {C} )}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}-1&a\\0&-1\end{pmatrix}}\in \mathrm {SL} (2,\mathbb {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d05f90d07d1fb2fd0697ca66c396db047a6e9dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.706ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}-1&a\\0&-1\end{pmatrix}}\in \mathrm {SL} (2,\mathbb {C} )}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f455a7f96d74aa94573d8e32da3b240ab0aa294f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" loading="lazy"> liegt nicht im Bild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}</annotation>

</semantics>

</td></tr></tbody></table>

Aus dem letzten Beispiel ist ersichtlich, dass die Exponentialabbildung für die Erzeugung von Lie-Gruppen (je nach Lie-Algebra) im Allgemeinen nicht surjektiv ist.

<div class="mw-heading mw-heading2"><h2 id="Lineare_Differentialgleichungen">Lineare Differentialgleichungen</h2></div>

Einer der Vorzüge des Matrixexponentials ist, dass man es benutzen kann, um Systeme von linearen <a href="Gew%C3%B6hnliche_Differentialgleichung" title="Gewöhnliche Differentialgleichung">gewöhnlichen Differentialgleichungen</a> zu lösen, die z. B. für das <a href="Zustandsraumdarstellung#Zustandsraummodell" title="Zustandsraumdarstellung">Zustandsraummodell</a> von dynamischen Übertragungssystemen verwendet werden. Aus Gleichung (1) oben folgt zum Beispiel, dass die Lösung des <a href="Anfangswertproblem" title="Anfangswertproblem">Anfangswertproblems</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}y(t)=Ay(t),\quad y(t_{0})=y_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/186e4e05d19a80fa0ee469485793b163169cef0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.849ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}y(t)=Ay(t),\quad y(t_{0})=y_{0}}" loading="lazy"></dd></dl>

mit einer quadratischen Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)=e^{A(t-t_{0})}y_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03c93cef519a1a42d192f902f48871073ebcae23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.222ex; height:3.343ex;" alt="{\displaystyle y(t)=e^{A(t-t_{0})}y_{0}}" loading="lazy"></dd></dl>

gegeben ist.

Das Matrixexponential kann auch zur Lösung der inhomogenen Gleichung

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}y(t)=Ay(t)+z(t),\quad y(t_{0})=y_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d72796b0a5568e53c3229631cf90a7c88364e30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.427ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}y(t)=Ay(t)+z(t),\quad y(t_{0})=y_{0}}" loading="lazy">,</dd></dl>

verwendet werden. Beispiele findet man unten im Kapitel <a href="#Anwendungen">Anwendungen</a>.

Für Differentialgleichungen der Form

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}y(t)=A(t)\,y(t),\quad y(t_{0})=y_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7147194c49a25cf9d38b3aa6c98ce82afdcce10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.885ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}y(t)=A(t)\,y(t),\quad y(t_{0})=y_{0}}" loading="lazy"></dd></dl>

mit nicht-konstantem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> gibt es im Allgemeinen keine geschlossenen Lösungen. Die Magnus-Reihe liefert jedoch eine allgemeine Lösung in Matrixschreibweise über die Matrix-Exponentialfunktion auch im Fall nicht-konstanter Koeffizienten (als unendliche Reihe des Exponenten).<a href="#cite_note-Blanes-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Berechnung_des_Matrixexponentials">Berechnung des Matrixexponentials</h2></div>

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

Die Exponentialfunktion der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbb {C} ^{n\times n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b3023de9ab6f59e511d7c8ae72d03b64bcecbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.745ex; height:2.343ex;" alt="{\displaystyle A\in \mathbb {C} ^{n\times n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)=\sum _{k=0}^{\infty }{\frac {A^{k}t^{k}}{k!}}=E+At+{\frac {A^{2}t^{2}}{2}}+{\frac {A^{3}t^{3}}{6}}+{\frac {A^{4}t^{4}}{24}}+\cdots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b6977259100231b998ed74848e5a3871e4b9e6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.345ex; height:7.009ex;" alt="{\displaystyle \exp(At)=\sum _{k=0}^{\infty }{\frac {A^{k}t^{k}}{k!}}=E+At+{\frac {A^{2}t^{2}}{2}}+{\frac {A^{3}t^{3}}{6}}+{\frac {A^{4}t^{4}}{24}}+\cdots }" loading="lazy"></dd></dl>

Hierbei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k!}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/949e312c9bf9a9a5e641c1db1b1d7c6f0425b536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k!}" loading="lazy"> die <a href="Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik)">Fakultät</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. Bei ausreichender Genauigkeit (Reihe ist <a href="Absolut_konvergente_Reihe" title="Absolut konvergente Reihe">absolut konvergent</a>) soll die Reihe bei einer endlichen Zahl an Berechnungsschritten abbrechen. Je größer die Einträge der Matrix sind, desto mehr Glieder der Reihe müssen aber berechnet werden (z. B. für die Lösung der linearen DGL für einen großen Zeitschritt). Um den Lösungsalgorithmus dahingehend zu verbessern, kann man die Einträge der Matrix mittels der Rechenregel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{At}=(e^{At/m})^{m}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25019cff0d0be58c4ba92862206929500032af40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.132ex; height:3.343ex;" alt="{\displaystyle e^{At}=(e^{At/m})^{m}}" loading="lazy"> elegant skalieren ("Scaling & Squaring"-Methode). Ist die (natürliche-) <a href="Matrixnorm" title="Matrixnorm">Matrixnorm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|At\|}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f8239952489fad18d962c0e4f7c88611d350ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle \|At\|}" loading="lazy"> nicht zu groß, kann die Berechnung der Reihe auch über die <a href="Pad%C3%A9-Approximation" title="Padé-Approximation">Padé-Approximation</a> erfolgen. Die Scaling & Squaring-Methode hat einen Aufwand der Größenordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff78e74de3bf7a5246829c66bc5acf0c2a94b67c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(n^{3})}" loading="lazy"> (im Wesentlichen <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikationen</a>). Der Faktor von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle n^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e9d1a52e455a7a5272a345b2697e35f1579b681.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{3}}" loading="lazy"> ist abhängig von den Skalierungsparametern sowie insbesondere von der Matrixnorm.

<div class="mw-heading mw-heading4"><h4 id="Nilpotenter_Fall">Nilpotenter Fall</h4></div>

Eine Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> ist <a href="Nilpotente_Matrix" title="Nilpotente Matrix">nilpotent</a>, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{q}=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dacb094a898d0805dcb98ee6896d8dc02d88ac4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.372ex; height:2.343ex;" alt="{\displaystyle N^{q}=0}" loading="lazy"> für eine geeignete natürliche Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> gilt. In diesem Fall bricht die Reihenentwicklung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{N}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a39ffd822d62657627c66c6e2b27f947b51cc81f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.676ex;" alt="{\displaystyle e^{N}}" loading="lazy"> nach einer endlichen Anzahl von Termen ab und das Matrixexponential kann als

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<annotation encoding="application/x-tex">{\displaystyle \exp(N)=E+N+{\frac {1}{2}}N^{2}+{\frac {1}{6}}N^{3}+\cdots +{\frac {1}{(q-1)!}}N^{q-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/297c421af7b0e066dd393ae78d63ffc8b6cea747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.217ex; height:6.009ex;" alt="{\displaystyle \exp(N)=E+N+{\frac {1}{2}}N^{2}+{\frac {1}{6}}N^{3}+\cdots +{\frac {1}{(q-1)!}}N^{q-1}}" loading="lazy"></dd></dl>

berechnet werden.

<div class="mw-heading mw-heading3"><h3 id="Diagonalisierung_der_Matrix">Diagonalisierung der Matrix</h3></div>

Ist die Matrix <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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<annotation encoding="application/x-tex">{\displaystyle D={\begin{pmatrix}a_{1}&0&\ldots &0\\0&a_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &a_{n}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/185975b92d61b06593b0850ffab92fa4f83bddf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:26.804ex; height:14.176ex;" alt="{\displaystyle D={\begin{pmatrix}a_{1}&0&\ldots &0\\0&a_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &a_{n}\end{pmatrix}}}" loading="lazy">,</dd></dl>

dann kann man ihr Exponential ermitteln, indem man die gewöhnliche Exponentialfunktion auf jeden Eintrag der <a href="Hauptdiagonale" title="Hauptdiagonale">Hauptdiagonalen</a> anwendet:

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<annotation encoding="application/x-tex">{\displaystyle e^{D}={\begin{pmatrix}e^{a_{1}}&0&\ldots &0\\0&e^{a_{2}}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &e^{a_{n}}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e79af8291eddadf2f3172b4f038be1c7826d48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.724ex; height:14.176ex;" alt="{\displaystyle e^{D}={\begin{pmatrix}e^{a_{1}}&0&\ldots &0\\0&e^{a_{2}}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &e^{a_{n}}\end{pmatrix}}}" loading="lazy">.</dd></dl>

Damit kann man auch das Exponential einer <a href="Diagonalisierbare_Matrix" title="Diagonalisierbare Matrix">diagonalisierbaren Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6507fc0061185accb5c1a67516910e9a20a1848.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.803ex; height:2.676ex;" alt="{\displaystyle A=VDV^{-1}}" loading="lazy"></dd></dl>

mit einer Diagonalmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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</math><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"> werden die zugehörige <a href="Eigenbasis" class="mw-redirect" title="Eigenbasis">Eigenbasis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> sowie die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><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"> <a href="Eigenwerte" class="mw-redirect" title="Eigenwerte">Eigenwerte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a896f82d292f2489a979c7a2c7a52561df77dd4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}" loading="lazy"> der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)=Ve^{Dt}\,V^{-1}=V\operatorname {diag} (e^{\lambda _{1}t},e^{\lambda _{2}t},\dotsc ,e^{\lambda _{n}t})\,V^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6c29957ceb99e1a4726260adcc2df756d5c7628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.164ex; height:3.176ex;" alt="{\displaystyle \exp(At)=Ve^{Dt}\,V^{-1}=V\operatorname {diag} (e^{\lambda _{1}t},e^{\lambda _{2}t},\dotsc ,e^{\lambda _{n}t})\,V^{-1}}" loading="lazy"></dd></dl>

mit der skalaren Exponentialfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto e^{\lambda _{i}t}}">

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<annotation encoding="application/x-tex">{\displaystyle t\mapsto e^{\lambda _{i}t}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6e3f954e4679eaa223949a06f2d9eaa4d31ddd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.946ex; height:2.676ex;" alt="{\displaystyle t\mapsto e^{\lambda _{i}t}}" loading="lazy">. Der Beweis folgt direkt aus der Taylor-Entwicklung der Exponentialfunktion.

Die Diagonalisierung der Matrix gehört, wie auch der <a href="QR-Algorithmus" title="QR-Algorithmus">QR-Algorithmus</a> oder die <a href="Jordansche_Normalform" title="Jordansche Normalform">Jordansche Normalform</a>, zu den Matrix-Zerlegungsmethoden zur Berechnung der Exponentialfunktion. Die Diagonalisierung und der QR-Algorithmus haben dabei jeweils einen Aufwand der Größenordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff78e74de3bf7a5246829c66bc5acf0c2a94b67c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(n^{3})}" loading="lazy"> und sind aber, im Vergleich zu Methoden auf Basis der Taylor-Entwicklung, unabhängig von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|At\|}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f8239952489fad18d962c0e4f7c88611d350ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle \|At\|}" loading="lazy">. Der wesentliche Berechnungsaufwand (hier: Bestimmung der Eigenwerte und <a href="Eigenvektoren" class="mw-redirect" title="Eigenvektoren">Eigenvektoren</a>) ist zudem unabhängig von der Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. Zur Lösung beispielsweise von linearen Differentialgleichungen für mehrere Zeitschritte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> muss dieser Arbeitsaufwand also nur einmalig erbracht werden. Die Berechnung der weiteren Zeitschritte erfolgt bei der Methode Diagonalisierung durch einfache Matrizenmultiplikation und bei dem QR-Algorithmus liegt der Aufwand in der Größenordnung von nur noch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{2})}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Beispiel_1">Beispiel 1</h4></div>

Es soll das Matrixexponential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{tA}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02f545c3cabe5d094a8fe06d56ee53fa3454f1f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.142ex; height:2.676ex;" alt="{\displaystyle e^{tA}}" loading="lazy"> für die folgende Matrix berechnet werden:

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}0&1\\a_{2,1}&a_{2,2}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e2413a62d4f01941810983547bba533a92fb252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.309ex; margin-bottom: -0.195ex; width:18.464ex; height:6.176ex;" alt="{\displaystyle A={\begin{pmatrix}0&1\\a_{2,1}&a_{2,2}\end{pmatrix}}}" loading="lazy">.</dd></dl>

Hierzu wird die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2\times 2)}">

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<annotation encoding="application/x-tex">{\displaystyle (2\times 2)}</annotation>

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</math><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">-Matrix <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><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"> zunächst mittels der Eigenwerte und den Eigenvektoren diagonalisiert. Mit der Diagonalmatrix <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><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"> und der Eigenbasis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)=Ve^{Dt}\,V^{-1}=V{\begin{pmatrix}e^{\lambda _{1}t}&0\\0&e^{\lambda _{2}t}\end{pmatrix}}V^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/331d90ae966951eeae1057c969a8b9b85dc9b265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.766ex; height:6.176ex;" alt="{\displaystyle \exp(At)=Ve^{Dt}\,V^{-1}=V{\begin{pmatrix}e^{\lambda _{1}t}&0\\0&e^{\lambda _{2}t}\end{pmatrix}}V^{-1}}" loading="lazy">.</dd></dl>

Die Eigenwerte werden aus dem <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristischen Polynom</a> bestimmt zu

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1,2}={\frac {a_{2,2}}{2}}\pm {\sqrt {\left({\frac {a_{2,2}}{2}}\right)^{2}+a_{2,1}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cf554e412ef0389e994abaf591642e59ad91eb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.984ex; height:6.176ex;" alt="{\displaystyle \lambda _{1,2}={\frac {a_{2,2}}{2}}\pm {\sqrt {\left({\frac {a_{2,2}}{2}}\right)^{2}+a_{2,1}}}}" loading="lazy">.</dd></dl>

Für die beiden Eigenvektoren bzw. die Eigenbasis gilt:

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<annotation encoding="application/x-tex">{\displaystyle V={\begin{pmatrix}1&1\\\lambda _{1}&\lambda _{2}\end{pmatrix}}~~~}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc069fdbd021a2a978b4f4aca0291c41dc17b200.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.942ex; height:6.176ex;" alt="{\displaystyle V={\begin{pmatrix}1&1\\\lambda _{1}&\lambda _{2}\end{pmatrix}}~~~}" loading="lazy">sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~V^{-1}={\frac {1}{\lambda _{2}-\lambda _{1}}}{\begin{pmatrix}\lambda _{2}&-1\\-\lambda _{1}&1\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle ~~V^{-1}={\frac {1}{\lambda _{2}-\lambda _{1}}}{\begin{pmatrix}\lambda _{2}&-1\\-\lambda _{1}&1\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b152a3e79a7d42c9ba80d3b370c99fecee9d29d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.688ex; height:6.176ex;" alt="{\displaystyle ~~V^{-1}={\frac {1}{\lambda _{2}-\lambda _{1}}}{\begin{pmatrix}\lambda _{2}&-1\\-\lambda _{1}&1\end{pmatrix}}}" loading="lazy"></dd></dl>

Einsetzen für die Matrix-Exponentialfunktion liefert schließlich

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)={\frac {1}{\lambda _{2}-\lambda _{1}}}{\begin{pmatrix}\lambda _{2}e^{\lambda _{1}t}-\lambda _{1}e^{\lambda _{2}t}&e^{\lambda _{2}t}-e^{\lambda _{1}t}\\\lambda _{1}\lambda _{2}(e^{\lambda _{1}t}-e^{\lambda _{2}t})&\lambda _{2}e^{\lambda _{2}t}-\lambda _{1}e^{\lambda _{1}t}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/598c340b8b287f8dc57327e0d191e850987de26a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.959ex; height:6.509ex;" alt="{\displaystyle \exp(At)={\frac {1}{\lambda _{2}-\lambda _{1}}}{\begin{pmatrix}\lambda _{2}e^{\lambda _{1}t}-\lambda _{1}e^{\lambda _{2}t}&e^{\lambda _{2}t}-e^{\lambda _{1}t}\\\lambda _{1}\lambda _{2}(e^{\lambda _{1}t}-e^{\lambda _{2}t})&\lambda _{2}e^{\lambda _{2}t}-\lambda _{1}e^{\lambda _{1}t}\end{pmatrix}}}" loading="lazy"></dd></dl>

als geschlossene analytische Lösung.<a href="#cite_note-Möller-2">[2]</a>

<div class="mw-heading mw-heading4"><h4 id="Beispiel_2">Beispiel 2</h4></div>

Die Matrix-Exponentialfunktion

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)=\exp \left[{\begin{pmatrix}3&-4\\4&-5\end{pmatrix}}t\right]={\begin{pmatrix}(1+4t)e^{-t}&-4te^{-t}\\4te^{-t}&(1-4t)e^{-t}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f8b964a2107f15c665ca517af32338d83f0f538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.116ex; height:6.509ex;" alt="{\displaystyle \exp(At)=\exp \left[{\begin{pmatrix}3&-4\\4&-5\end{pmatrix}}t\right]={\begin{pmatrix}(1+4t)e^{-t}&-4te^{-t}\\4te^{-t}&(1-4t)e^{-t}\end{pmatrix}}}" loading="lazy"></dd></dl>

kann explizit berechnet werden, jedoch ist die Matrix <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><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"> selbst nicht diagonalisierbar. Die Matrix besitzt die beiden Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1,2}=-1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10f3156ff710ae5aab00b96db2b98bca8f2def5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.758ex; height:2.843ex;" alt="{\displaystyle \lambda _{1,2}=-1}" loading="lazy">. Obwohl also der Eigenwert die algebraische Vielfachheit 2 hat, existiert nur ein linear unabhängiger Eigenvektor. Die Basis aus den Eigenvektoren

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<annotation encoding="application/x-tex">{\displaystyle V={\begin{pmatrix}a_{2,1}&a_{2,1}\\\lambda _{1}-a_{1,1}&\lambda _{2}-a_{1,1}\end{pmatrix}}={\begin{pmatrix}-4&-4\\-4&-4\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b136a4e56d9e7826837410f453cb43056cf378ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.542ex; height:6.509ex;" alt="{\displaystyle V={\begin{pmatrix}a_{2,1}&a_{2,1}\\\lambda _{1}-a_{1,1}&\lambda _{2}-a_{1,1}\end{pmatrix}}={\begin{pmatrix}-4&-4\\-4&-4\end{pmatrix}}}" loading="lazy"></dd></dl>

ist nicht <a href="Invertierbare_Matrix" class="mw-redirect" title="Invertierbare Matrix">invertierbar</a>. Die <a href="Diskriminante" title="Diskriminante">Diskriminante</a> des charakteristischen Polynoms

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<annotation encoding="application/x-tex">{\displaystyle D_{2}=\left({\frac {a_{1,1}-a_{2,2}}{2}}\right)^{2}+a_{1,2}\,a_{2,1}=\left({\frac {3-(-5)}{2}}\right)^{2}-16}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5519e0a77207e014ee091849c0628375092b55d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.068ex; height:6.676ex;" alt="{\displaystyle D_{2}=\left({\frac {a_{1,1}-a_{2,2}}{2}}\right)^{2}+a_{1,2}\,a_{2,1}=\left({\frac {3-(-5)}{2}}\right)^{2}-16}" loading="lazy"></dd></dl>

wird dabei immer null. In diesem Fall, also wenn gleiche Eigenwerte bzw. Eigenvektoren vorkommen, kann formell die Jordansche Normalform zur Transformation verwendet werden.

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

Zerfällt das <a href="Minimalpolynom_(Lineare_Algebra)" title="Minimalpolynom (Lineare Algebra)">Minimalpolynom</a> (bzw. das <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristische Polynom</a>) der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in <a href="Linearfaktor" class="mw-redirect" title="Linearfaktor">Linearfaktoren</a> (über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> ist das stets der Fall), dann kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8422a31749136baaa162e847b6e0e46772bede3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.726ex; height:2.343ex;" alt="{\displaystyle X=A+N}" loading="lazy"></dd></dl>

zerlegt werden, wobei

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</math><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"> mit <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> kommutiert (d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AN=NA}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c23c2fbf2cdf0c4d814da3332296014b6d48a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.712ex; height:2.176ex;" alt="{\displaystyle AN=NA}" loading="lazy">).</li></ul>

Damit kann man das Exponential von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> berechnen, indem man es auf die vorgenannten Fälle reduziert: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X}=e^{A+N}=e^{A}e^{N}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{X}=e^{A+N}=e^{A}e^{N}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0dd7a61dae81c990b40e6bbad25861a5645e2d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.522ex; height:2.676ex;" alt="{\displaystyle e^{X}=e^{A+N}=e^{A}e^{N}}" loading="lazy">. Im letzten Schritt benötigt man die Kommutativität von <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><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"> und <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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<div class="mw-heading mw-heading3"><h3 id="Verwendung_der_jordanschen_Normalform">Verwendung der jordanschen Normalform</h3></div>

Eine weitere Methode ist die Verwendung der <a href="Jordansche_Normalform" title="Jordansche Normalform">jordanschen Normalform</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, wobei auch die Splitting-Methode zum Einsatz kommt. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> die jordansche Normalform von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> mit der <a href="Basiswechsel_(Vektorraum)" title="Basiswechsel (Vektorraum)">Basiswechselmatrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">, dann gilt

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<annotation encoding="application/x-tex">{\displaystyle e^{X}=Pe^{J}P^{-1}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbe21c69faf70ed24c5fae73ccf5398dd1c012b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.717ex; height:2.676ex;" alt="{\displaystyle e^{X}=Pe^{J}P^{-1}.}" loading="lazy"></dd></dl>

Wegen

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<annotation encoding="application/x-tex">{\displaystyle J=J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8cf188ef7644c9ea81b79ed2887e20b1034c30f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.439ex; height:3.009ex;" alt="{\displaystyle J=J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n})}" loading="lazy"></dd></dl>

gilt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{J}&=\exp {\big (}J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n}){\big )}\\&=\exp {\big (}J_{a_{1}}(\lambda _{1}){\big )}\oplus \exp {\big (}J_{a_{2}}(\lambda _{2}){\big )}\oplus \cdots \oplus \exp {\big (}J_{a_{k}}(\lambda _{k}){\big )}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13fe502f55903221f7f0718a320f3d86f0b0b404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:58.695ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}e^{J}&=\exp {\big (}J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n}){\big )}\\&=\exp {\big (}J_{a_{1}}(\lambda _{1}){\big )}\oplus \exp {\big (}J_{a_{2}}(\lambda _{2}){\big )}\oplus \cdots \oplus \exp {\big (}J_{a_{k}}(\lambda _{k}){\big )}.\end{aligned}}}" loading="lazy"></dd></dl>

Daher muss man nur das Exponential eines Jordan-Blocks kennen. Nun ist jeder Jordan-Block von der Form

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<annotation encoding="application/x-tex">{\displaystyle J_{a}(\lambda )=\lambda \,E+N,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3c16fe4d4904e02642a782123db18b11d6618af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.724ex; height:2.843ex;" alt="{\displaystyle J_{a}(\lambda )=\lambda \,E+N,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle e^{\lambda E+N}=e^{\lambda }e^{N}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6570a5c9d0a9d62a82ec829f66690a5efc9709ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.062ex; height:2.676ex;" alt="{\displaystyle e^{\lambda E+N}=e^{\lambda }e^{N}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Beispiel">Beispiel</h4></div>

Man betrachte die Matrix

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<annotation encoding="application/x-tex">{\displaystyle B={\begin{pmatrix}21&17&6\\-5&-1&-6\\4&4&16\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57b0b0ecb0c11bf1f944a239fb6b9b78158c6412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.238ex; height:9.176ex;" alt="{\displaystyle B={\begin{pmatrix}21&17&6\\-5&-1&-6\\4&4&16\end{pmatrix}}}" loading="lazy">,</dd></dl>

welche die <a href="Jordansche_Normalform" title="Jordansche Normalform">jordansche Normalform</a>

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<annotation encoding="application/x-tex">{\displaystyle J=P^{-1}BP={\begin{pmatrix}4&0&0\\0&16&1\\0&0&16\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8696921fa9807a8776c192753ba1291ad426d446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.608ex; height:9.176ex;" alt="{\displaystyle J=P^{-1}BP={\begin{pmatrix}4&0&0\\0&16&1\\0&0&16\end{pmatrix}}}" loading="lazy"></dd></dl>

mit der Übergangsmatrix

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<annotation encoding="application/x-tex">{\displaystyle P={\begin{pmatrix}-1&1&{\tfrac {5}{8}}\\1&-1&-{\tfrac {1}{8}}\\0&2&0\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/723b68840f254bd5893e08e8c6f62d1f0e5e10fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:23.715ex; height:10.843ex;" alt="{\displaystyle P={\begin{pmatrix}-1&1&{\tfrac {5}{8}}\\1&-1&-{\tfrac {1}{8}}\\0&2&0\end{pmatrix}}}" loading="lazy"></dd></dl>

hat. Dann gelten

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<annotation encoding="application/x-tex">{\displaystyle J=J_{1}(4)\oplus J_{2}(16)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88f45865d147222d8b16ef38760e91c30aded3dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.205ex; height:2.843ex;" alt="{\displaystyle J=J_{1}(4)\oplus J_{2}(16)}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle e^{B}=Pe^{J}P^{-1}=P(e^{J_{2}(16)}\oplus e^{J_{1}(4)})P^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167429a5013effc4514494cd808c7a876b03caa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.709ex; height:3.343ex;" alt="{\displaystyle e^{B}=Pe^{J}P^{-1}=P(e^{J_{2}(16)}\oplus e^{J_{1}(4)})P^{-1}}" loading="lazy">.</dd></dl>

Somit ist

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<annotation encoding="application/x-tex">{\displaystyle \exp \left[{\begin{pmatrix}16&0\\0&16\end{pmatrix}}+{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]=e^{16}E\cdot \left[{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+{\begin{pmatrix}0&1\\0&0\end{pmatrix}}+{1 \over 2!}{\begin{pmatrix}0&0\\0&0\end{pmatrix}}+\cdots \right]={\begin{pmatrix}e^{16}&e^{16}\\0&e^{16}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cd9faeed0dabbd83bbed4254106d728fb196fca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:96.644ex; height:6.176ex;" alt="{\displaystyle \exp \left[{\begin{pmatrix}16&0\\0&16\end{pmatrix}}+{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]=e^{16}E\cdot \left[{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+{\begin{pmatrix}0&1\\0&0\end{pmatrix}}+{1 \over 2!}{\begin{pmatrix}0&0\\0&0\end{pmatrix}}+\cdots \right]={\begin{pmatrix}e^{16}&e^{16}\\0&e^{16}\end{pmatrix}}}" loading="lazy">.</dd></dl>

Das Exponential einer 1×1-Matrix ist trivial. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{J_{1}(4)}=e^{4}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{J_{1}(4)}=e^{4}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a85c896294be6014b11bb302f68db9d54e99e747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.397ex; height:2.843ex;" alt="{\displaystyle e^{J_{1}(4)}=e^{4}}" loading="lazy"> folgt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\exp(B)&=P\exp(J)P^{-1}=P{\begin{pmatrix}e^{4}&0&0\\0&e^{16}&e^{16}\\0&0&e^{16}\end{pmatrix}}P^{-1}\\[6pt]&={1 \over 4}{\begin{pmatrix}13e^{16}-e^{4}&13e^{16}-5e^{4}&2e^{16}-2e^{4}\\-9e^{16}+e^{4}&-9e^{16}+5e^{4}&-2e^{16}+2e^{4}\\16e^{16}&16e^{16}&4e^{16}\end{pmatrix}}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7db42b03f2854d66f3627295c0b38c7c34505421.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:58.135ex; height:20.843ex;" alt="{\displaystyle {\begin{aligned}\exp(B)&=P\exp(J)P^{-1}=P{\begin{pmatrix}e^{4}&0&0\\0&e^{16}&e^{16}\\0&0&e^{16}\end{pmatrix}}P^{-1}\\[6pt]&={1 \over 4}{\begin{pmatrix}13e^{16}-e^{4}&13e^{16}-5e^{4}&2e^{16}-2e^{4}\\-9e^{16}+e^{4}&-9e^{16}+5e^{4}&-2e^{16}+2e^{4}\\16e^{16}&16e^{16}&4e^{16}\end{pmatrix}}.\end{aligned}}}" loading="lazy"></dd></dl>

Die jordansche Normalform und daraus das Exponential zu berechnen, ist auf diesem Weg sehr mühsam. Meist reicht es, die Wirkung der Exponential-Matrix auf einige Vektoren zu berechnen.

<div class="mw-heading mw-heading4"><h4 id="Numerische_Berechnung">Numerische Berechnung</h4></div>

Die Jordan-Normalform-Zerlegung ist <a href="Stabilit%C3%A4t_(Numerik)" title="Stabilität (Numerik)">numerisch instabil</a>, da aufgrund der Gleitkommaarithmetik <a href="Rundungsfehler" title="Rundungsfehler">Rundungsfehler</a> in die Eigenwerte eingeführt werden, die eine Gruppierung der Eigenwerte in Gruppen identischer Eigenwerte unmöglich macht. Daher werden in der Numerik andere Techniken zur Berechnung des Matrixexponentials verwendet. Zu den effektivsten verfügbaren Algorithmen gehören die <a href="Pad%C3%A9-Approximation" title="Padé-Approximation">Padé-Approximation</a> mit Skalieren und Quadrieren (s. <a href="#Taylor-Reihe">Berechnung mittels Taylorreihe</a>) oder die Matrix-Zerlegungsmethoden wie die <a href="#Diagonalisierung_der_Matrix">Diagonalisierung der Matrix</a>. Bei großen Matrizen kann der Rechenaufwand auch reduziert werden, indem <a href="Krylowraum" title="Krylowraum">Krylowräume</a> verwendet werden, deren Basisvektoren mit dem <a href="Arnoldi-Verfahren" title="Arnoldi-Verfahren">Arnoldi-Verfahren</a> orthogonalisiert worden sind.

<div class="mw-heading mw-heading3"><h3 id="Explizite_Formeln">Explizite Formeln</h3></div>

Mittels der Diagonalisierung über die Eigenwerte und Eigenvektoren

ist die Darstellung der Matrix-Exponentialfunktion einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8a0e3400ffb97d67c00267ed50cddfe824cbe80.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 2\times 2}" loading="lazy">-Matrix auch als explizite Formel möglich.

Insbesondere der Umweg über die teils komplexen Eigenwerte ist damit nicht mehr

notwendig:<a href="#cite_note-Möller2-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)={\begin{pmatrix}{\frac {a_{1,1}-a_{2,2}}{2}}f(t)+f'(t)&a_{1,2}f(t)\\a_{2,1}f(t)&{\frac {a_{2,2}-a_{1,1}}{2}}f(t)+f'(t)\end{pmatrix}}e^{{\frac {a_{1,1}+a_{2,2}}{2}}t}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6ac80fce18eae1a713f91137337fcc86c1920b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:65.091ex; height:8.509ex;" alt="{\displaystyle \exp(At)={\begin{pmatrix}{\frac {a_{1,1}-a_{2,2}}{2}}f(t)+f'(t)&a_{1,2}f(t)\\a_{2,1}f(t)&{\frac {a_{2,2}-a_{1,1}}{2}}f(t)+f'(t)\end{pmatrix}}e^{{\frac {a_{1,1}+a_{2,2}}{2}}t}}" loading="lazy"></dd></dl>

Für die Hilfsfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"> gilt in Abhängigkeit der reellen, komplexen, oder gleichen

Eigenwerte (anhand der <a href="Diskriminante" title="Diskriminante">Diskriminante</a> <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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<annotation encoding="application/x-tex">{\displaystyle D=\left({\frac {a_{1,1}-a_{2,2}}{2}}\right)^{2}+a_{1,2}a_{2,1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7424ae3bea239920924ba2804a7154903db761ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.269ex; height:6.509ex;" alt="{\displaystyle D=\left({\frac {a_{1,1}-a_{2,2}}{2}}\right)^{2}+a_{1,2}a_{2,1}}" loading="lazy"></th>

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

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<annotation encoding="application/x-tex">{\displaystyle f'(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2b503f27f6df6a6fbc77596c078bb6a25311557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.654ex; height:3.009ex;" alt="{\displaystyle f'(t)}" loading="lazy">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b590bbbcf66fd5a62c56490804db67a445a6548b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle >0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sinh \left({\sqrt {D}}t\right)}{\sqrt {D}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe2dabccd79ed5945cfbbe1d6da483a9b5bb5d29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.814ex; height:7.176ex;" alt="{\displaystyle {\frac {\sinh \left({\sqrt {D}}t\right)}{\sqrt {D}}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \cosh \left({\sqrt {D}}t\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c17d1afadce5c1dc8b6b4b40471a077c74805aa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.233ex; height:3.343ex;" alt="{\displaystyle \cosh \left({\sqrt {D}}t\right)}" loading="lazy">

</td></tr>

<tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5421042136c65f8858e00b79187c0f1742ee919.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle <0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin \left({\sqrt {|D|}}t\right)}{\sqrt {|D|}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abfc9c3647ddae6ed40aed0baa6f372890b7efd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.203ex; height:7.509ex;" alt="{\displaystyle {\frac {\sin \left({\sqrt {|D|}}t\right)}{\sqrt {|D|}}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \cos \left({\sqrt {|D|}}t\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/105ace3a6301657aba5b91e5028449f085088c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.268ex; height:4.843ex;" alt="{\displaystyle \cos \left({\sqrt {|D|}}t\right)}" loading="lazy">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dc9e66de468806365c20e32e83456cc526ce29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle =0}" loading="lazy">

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

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

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</math><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">

</td></tr></tbody></table>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Putzer-Algorithmus" title="Putzer-Algorithmus">Putzer-Algorithmus</a></div>

Eine weitere Möglichkeit, das Matrixexpontial zu berechnen, ist der Putzer-Algorithmus. Dabei definiert man bei gegebener Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbb {C} ^{n\times n}}">

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/592bced0c39b10fc90e74c6a66223abfbfb029de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }" loading="lazy"> rekursiv stetig-differenzierbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{k}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01084a31964201514f3e6bd0136989e11ea6e58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:2.009ex;" alt="{\displaystyle p_{k}}" loading="lazy"> und Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43175c381ee6ba2694520af1f1a26c676a2726ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}" loading="lazy">, so dass gilt:

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<annotation encoding="application/x-tex">{\displaystyle \exp(At)=\sum _{k=1}^{n}p_{k}(t)M_{k-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba29f391be1a64424f63de0946573969131de066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.135ex; height:6.843ex;" alt="{\displaystyle \exp(At)=\sum _{k=1}^{n}p_{k}(t)M_{k-1}}" loading="lazy"></dd></dl>

Die Lösung des Matrixexponentials einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\times n)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0e369b52ee16c33d83f7cd0eb0f562fd91b7f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.439ex; height:2.843ex;" alt="{\displaystyle (n\times n)}" loading="lazy">-Matrix wird hierbei als Polynom erhalten. Die Berechnung hat dabei einen Aufwand der Größenordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{4})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{4})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25dcc2a2a37c4bfafb26542489d56d0e6adacf8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(n^{4})}" loading="lazy"> (Berechnung der Eigenwerte sowie insbesondere <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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<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Homogene_lineare_Differentialgleichungen">Homogene lineare Differentialgleichungen</h3></div>

Das Matrixexponential kann für die Lösung eines Systems von <a href="Lineare_Differentialgleichung" class="mw-redirect" title="Lineare Differentialgleichung">linearen Differentialgleichungen</a> verwendet werden. Eine Differentialgleichung der Form

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<annotation encoding="application/x-tex">{\displaystyle y'=Cy}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75e11f32a61c0f120685851c7e3f4740d2e02e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.865ex; height:2.843ex;" alt="{\displaystyle y'=Cy}" loading="lazy"></dd></dl>

hat die Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{Ct}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d60a44c9b4736f13703e5b9a901fb23c09ffbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.158ex; height:2.676ex;" alt="{\displaystyle e^{Ct}}" loading="lazy">. Wenn man den Vektor

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} (t)={\begin{pmatrix}y_{1}(t)\\\vdots \\y_{n}(t)\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9a4e7c6ca704ada74e422e997b6c2db3a23fcc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:16.983ex; height:10.843ex;" alt="{\displaystyle \mathbf {y} (t)={\begin{pmatrix}y_{1}(t)\\\vdots \\y_{n}(t)\end{pmatrix}}}" loading="lazy"></dd></dl>

betrachtet, dann kann man ein System von gekoppelten linearen Differentialgleichungen betrachten als

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} '(t)=A\mathbf {y} (t)+\mathbf {b} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d617dcb7d71f07379e933060275d63c4256a18e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.972ex; height:3.009ex;" alt="{\displaystyle \mathbf {y} '(t)=A\mathbf {y} (t)+\mathbf {b} }" loading="lazy">.</dd></dl>

Wenn man den Integrationsfaktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-tA}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7aedb5c810b43b0d2d0d17ed97c3368fd5632abe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.421ex; height:2.676ex;" alt="{\displaystyle e^{-tA}}" loading="lazy"> ansetzt und auf beiden Seiten multipliziert, erhält man

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<annotation encoding="application/x-tex">{\displaystyle e^{-tA}\mathbf {y} '(t)-e^{-tA}A\mathbf {y} =e^{-tA}\mathbf {b} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/363171b849bf111aa96062ac4f9bfaafbd765a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.584ex; height:3.176ex;" alt="{\displaystyle e^{-tA}\mathbf {y} '(t)-e^{-tA}A\mathbf {y} =e^{-tA}\mathbf {b} }" loading="lazy">,</dd></dl>

also

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(e^{-tA}\mathbf {y} )=e^{-tA}\mathbf {b} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e29d00a6f38099aba654bc653f4c91b9b917156d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.613ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(e^{-tA}\mathbf {y} )=e^{-tA}\mathbf {b} }" loading="lazy">.</dd></dl>

Wenn man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{tA}}">

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

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<div class="mw-heading mw-heading4"><h4 id="Beispiel_(homogen)">Beispiel (homogen)</h4></div>

Gegeben sei das folgende Differentialgleichungssystem

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{1}'&=3y_{1}-y_{2},\\y_{2}'&=y_{1}+y_{2}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96e29e3f951a2e2ff9b0b9c9fd2f8a337f979136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.08ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}y_{1}'&=3y_{1}-y_{2},\\y_{2}'&=y_{1}+y_{2}.\end{aligned}}}" loading="lazy"></dd></dl>

Es lässt sich schreiben als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'(t)=Ay(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y'(t)=Ay(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dea9fce54fc86aaad78ff6a2fa35cb7872921175.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.14ex; height:3.009ex;" alt="{\displaystyle y'(t)=Ay(t)}" loading="lazy"> mit der Koeffizientenmatrix

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}3&-1\\1&1\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e77225a50e6214430e8fe010602aa59b74cca95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.47ex; height:6.176ex;" alt="{\displaystyle A={\begin{pmatrix}3&-1\\1&1\end{pmatrix}}}" loading="lazy">.</dd></dl>

Damit ergibt sich das zugehörige Matrixexponential zu

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<annotation encoding="application/x-tex">{\displaystyle e^{tA}=e^{2t}{\begin{pmatrix}1+t&-t\\t&1-t\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6b855d3215ecbbc9362e8aaf25a8703ec389757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.152ex; height:6.176ex;" alt="{\displaystyle e^{tA}=e^{2t}{\begin{pmatrix}1+t&-t\\t&1-t\end{pmatrix}}}" loading="lazy">.</dd></dl>

Als allgemeine Lösung des Differentialgleichungssystems erhält man somit

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}y_{1}(t)\\y_{2}(t)\end{pmatrix}}=C_{1}e^{2t}{\begin{pmatrix}1+t\\t\end{pmatrix}}+C_{2}e^{2t}{\begin{pmatrix}-t\\1-t\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf174b381cf10273128622c8a742334eb7f8bbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.879ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}y_{1}(t)\\y_{2}(t)\end{pmatrix}}=C_{1}e^{2t}{\begin{pmatrix}1+t\\t\end{pmatrix}}+C_{2}e^{2t}{\begin{pmatrix}-t\\1-t\end{pmatrix}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Inhomogener_Fall_–_Variation_der_Konstanten">Inhomogener Fall – Variation der Konstanten</h3></div>

Für den inhomogenen Fall kann man die Methode der <a href="Variation_der_Konstanten" title="Variation der Konstanten">Variation der Konstanten</a> benutzen. Es wird eine Lösung der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{p}(t)=e^{tA}\mathbf {z} (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}(t)=e^{tA}\mathbf {z} (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e67f80a198e10fe471ce8ba42439c3e335c48a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.196ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{p}(t)=e^{tA}\mathbf {z} (t)}" loading="lazy"> gesucht:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}'(t)=(e^{tA})'\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)=Ae^{tA}\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)=A\mathbf {y} _{p}(t)+e^{tA}\mathbf {z} '(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fca2a365d370234df608ede24aa43396f31398da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:70.984ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{p}'(t)=(e^{tA})'\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)=Ae^{tA}\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)=A\mathbf {y} _{p}(t)+e^{tA}\mathbf {z} '(t)}" loading="lazy"></dd></dl>

Um die Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d5e97c4bd8e66c546e632fbc1c04f109234d0c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.47ex; height:2.509ex;" alt="{\displaystyle \mathbf {y} _{p}}" loading="lazy"> zu ermitteln, fordert man

<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{tA}\mathbf {z} '(t)=\mathbf {b} (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce73caf9d7f5ae8b8c57cb0912cce2b59c89d61a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.896ex; height:3.176ex;" alt="{\displaystyle e^{tA}\mathbf {z} '(t)=\mathbf {b} (t)}" loading="lazy">,</dd></dl>

also

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} '(t)=e^{-tA}\mathbf {b} (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0ac209e32dd4c07f02dd225464be92c845ac339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.175ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} '(t)=e^{-tA}\mathbf {b} (t)}" loading="lazy"></dd></dl>

und daher

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} (t)=\mathbf {z} (t_{0})+\int _{t_{0}}^{t}e^{-uA}\mathbf {b} (u)\,du}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecd0681acec0f325f638060e325d9a4f5fc74f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.668ex; height:6.509ex;" alt="{\displaystyle \mathbf {z} (t)=\mathbf {z} (t_{0})+\int _{t_{0}}^{t}e^{-uA}\mathbf {b} (u)\,du}" loading="lazy">.</dd></dl>

Damit ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}(t)=e^{tA}\mathbf {z} (t_{0})+e^{tA}\int _{t_{0}}^{t}e^{-uA}\mathbf {b} (u)\,du=e^{(t-t_{0})A}\,\mathbf {y} _{0}+\int _{t_{0}}^{t}e^{(t-u)A}\mathbf {b} (u)\,du}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2fb5c1e4c3db5dc668ab0f2230c39fd93d93e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.412ex; height:6.509ex;" alt="{\displaystyle \mathbf {y} _{p}(t)=e^{tA}\mathbf {z} (t_{0})+e^{tA}\int _{t_{0}}^{t}e^{-uA}\mathbf {b} (u)\,du=e^{(t-t_{0})A}\,\mathbf {y} _{0}+\int _{t_{0}}^{t}e^{(t-u)A}\mathbf {b} (u)\,du}" loading="lazy">,</dd></dl>

wobei die Anfangsbedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{p}(t_{0})=\mathbf {y} _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}(t_{0})=\mathbf {y} _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4ab8276ad2750b6e83076d439f54085340dfe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.737ex; height:3.176ex;" alt="{\displaystyle \mathbf {y} _{p}(t_{0})=\mathbf {y} _{0}}" loading="lazy"> angenommen worden ist.

<div class="mw-heading mw-heading4"><h4 id="Beispiel_(inhomogen)">Beispiel (inhomogen)</h4></div>

Gegeben sei das Differentialgleichungssystem

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{1}'&=3y_{1}-y_{2}-2e^{t},\\y_{2}'&=y_{1}+y_{2}-e^{t}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcb7d30b54a264451e010e395d3c48b1abd8821a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.992ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}y_{1}'&=3y_{1}-y_{2}-2e^{t},\\y_{2}'&=y_{1}+y_{2}-e^{t}.\end{aligned}}}" loading="lazy"></dd></dl>

Mit der Matrix <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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<annotation encoding="application/x-tex">{\displaystyle y'(t)=Ay(t)+b(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed24b52e978b94ba0fec4c936ebbcb73472a1dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.627ex; height:3.009ex;" alt="{\displaystyle y'(t)=Ay(t)+b(t)}" loading="lazy"></dd></dl>

mit

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<annotation encoding="application/x-tex">{\displaystyle b(t)=e^{t}{\begin{pmatrix}-2\\-1\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cd03a58a93db65c44177505d5cfef618d50a85c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.798ex; height:6.176ex;" alt="{\displaystyle b(t)=e^{t}{\begin{pmatrix}-2\\-1\end{pmatrix}}}" loading="lazy">.</dd></dl>

Die allgemeine Lösung der homogenen Gleichung wurde bereits oben berechnet. Die Summe aus homogenen und speziellen Lösungen ergibt die Lösung für das inhomogene Problem. Man muss jetzt nur noch eine spezielle Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{p}(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9a760f8f115deba534d8731452d7f0ddacbac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.847ex; height:3.009ex;" alt="{\displaystyle y_{p}(t)}" loading="lazy"> finden (über die Variation der Konstanten). Von der Gleichung oben erhält man:

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<annotation encoding="application/x-tex">{\displaystyle y_{p}(t)=e^{tA}\int _{0}^{t}e^{-uA}b(u)\,du}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7a12a818f16fe92edfb5aad3846c14d4c08f785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.988ex; height:6.176ex;" alt="{\displaystyle y_{p}(t)=e^{tA}\int _{0}^{t}e^{-uA}b(u)\,du}" loading="lazy">,</dd></dl>

also

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<annotation encoding="application/x-tex">{\displaystyle y_{p}(t)=e^{2t}{\begin{pmatrix}1+t&-t\\t&1-t\end{pmatrix}}\int _{0}^{t}e^{-2u}{\begin{pmatrix}1-u&u\\-u&1+u\end{pmatrix}}{\begin{pmatrix}-2e^{u}\\-e^{u}\end{pmatrix}}\,du={\begin{pmatrix}-e^{2t}(1+t)+e^{t}\\-te^{2t}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e14730ccb5960df91b1fe6e227514c4ea5400185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:87.983ex; height:6.509ex;" alt="{\displaystyle y_{p}(t)=e^{2t}{\begin{pmatrix}1+t&-t\\t&1-t\end{pmatrix}}\int _{0}^{t}e^{-2u}{\begin{pmatrix}1-u&u\\-u&1+u\end{pmatrix}}{\begin{pmatrix}-2e^{u}\\-e^{u}\end{pmatrix}}\,du={\begin{pmatrix}-e^{2t}(1+t)+e^{t}\\-te^{2t}\end{pmatrix}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrizen</a></li>

<li><a href="Matrixlogarithmus" title="Matrixlogarithmus">Matrixlogarithmus</a></li>

<li><a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a></li>

<li><a href="Exponentialabbildung" title="Exponentialabbildung">Exponentialabbildung</a></li></ul>

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

<ul><li>Roger A. Horn, Charles R. Johnson: Topics in Matrix Analysis. Cambridge University Press, 1991, ISBN 0-521-46713-6 (englisch).</li>

<li>Cleve Moler, Charles F. Van Loan: <cite style="font-style:italic">Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later</cite>. In: <cite style="font-style:italic">SIAM Review</cite>. Band 45, Nr. 1, 2003, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%221095-7200%22&key=cql">1095-7200</a>, S. 1–49, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137/S00361445024180">10.1137/S00361445024180</a> (<a rel="nofollow" class="external text" href="http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf">cornell.edu</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Matrixexponential&rft.atitle=Nineteen+Dubious+Ways+to+Compute+the+Exponential+of+a+Matrix%2C+Twenty-Five+Years+Later&rft.au=Cleve+Moler%2C+Charles+F.+Van+Loan&rft.date=2003&rft.doi=10.1137%2FS00361445024180&rft.genre=journal&rft.issn=1095-7200&rft.issue=1&rft.jtitle=SIAM+Review&rft.pages=1-49&rft.volume=45" style="display:none"> </li>

<li>V. I. Arnolʹd: Gewöhnliche Differentialgleichungen. Springer-Verlag, Berlin / New York 1980, ISBN 3-540-09216-1.</li></ul>

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

<li id="cite_note-Blanes-1"><a href="#cite_ref-Blanes_1-0">↑</a> S. Blanes, F. Casas, J. A. Oteo, J. Ros: The Magnus expansion and some of its applications. (= Physics Reports. Band 470). Cornell University Library, 2009, <a href="OCLC" title="OCLC">OCLC</a> <a rel="nofollow" class="external text" href="https://worldcat.org/oclc/635162561">635162561</a>.

</li>

<li id="cite_note-Möller-2"><a href="#cite_ref-Möller_2-0">↑</a> T. Möller: Symbolic mathematics-based simulation of cylinder spaces for regenerative gas cycles. In: Int J Energy Environ Eng. Springer Berlin/Heidelberg, Feb. 2015. <a rel="nofollow" class="external free" href="https://link.springer.com/article/10.1007/s40095-015-0163-3">https://link.springer.com/article/10.1007/s40095-015-0163-3</a>

</li>

<li id="cite_note-Möller2-3"><a href="#cite_ref-Möller2_3-0">↑</a> T. Möller: Simulation und konstruktive Optimierung der Wärmeübertrager regenerativer Gaskreisprozesse. Shaker Verlag Düren, Aug. 2022. <a rel="nofollow" class="external free" href="https://www.shaker.de/shop/978-3-8440-8706-2">https://www.shaker.de/shop/978-3-8440-8706-2</a>

</li>

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