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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Verallgemeinertes Eigenwertproblem</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>verallgemeinerte <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenwertproblem</a></b> ist eine Problemstellung der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Das Problem, zu vorgegebenen Matrizen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in \mathbb {C} ^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B\in \mathbb {C} ^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fab8e400f8f9009373e9b564a9d5b716b72a797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.543ex; height:2.676ex;" alt="{\displaystyle A,B\in \mathbb {C} ^{n\times n}}" loading="lazy"></span> gewisse Zahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11a6d1585381827bdf73529c2a418bc14098567c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {C} }" loading="lazy"></span> und Vektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {C} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {C} ^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d487cadb233e86b6a5c373836a6fe5a8d22c55c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {C} ^{n}}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy"></span> zu bestimmen, sodass
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=\lambda Bx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mi>x</mi>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mi>B</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Ax=\lambda Bx}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74fef2c4c90beb0b31d7f446d6f5bdbbc1c1ca78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.62ex; height:2.176ex;" alt="{\displaystyle Ax=\lambda Bx}" loading="lazy"></span></dd></dl>
<p>gilt, wird in Abgrenzung zum Eigenwertproblem als <i>verallgemeinertes Eigenwertproblem</i> bezeichnet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Lösungsverfahren"><span id="L.C3.B6sungsverfahren"></span>Lösungsverfahren</h2></div>
<p>Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> <a href="Regul%C3%A4re_Matrix" title="Reguläre Matrix">regulär</a>, so lässt sich das verallgemeinerte Eigenwertproblem auf das gewöhnliche Eigenwertproblem
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{-1}Ax=\lambda x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
<mi>x</mi>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B^{-1}Ax=\lambda x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b01c9c1b6fdcc913401a87e43e4add630bf2da00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.953ex; height:2.676ex;" alt="{\displaystyle B^{-1}Ax=\lambda x}" loading="lazy"></span></dd></dl>
<p>zurückführen. Dieser Lösungsansatz ist aber i. A. nur von theoretischer Bedeutung, da die Berechnung einer <a href="Inverse_Matrix" title="Inverse Matrix">inversen Matrix</a> numerisch oft nicht möglich oder sehr unpraktisch ist.
Oftmals lassen sich aus der Aufgabenstellung schon gewisse Informationen über die betrachteten Matrizen sammeln, welche die Berechnung dann vereinfachen können. Sind z. B. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"></span> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> außerdem <a href="Definitheit" title="Definitheit">positiv definit</a>, so lässt sich die Berechnung wesentlich vereinfachen: Die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> lässt sich mittels der <a href="Cholesky-Zerlegung" title="Cholesky-Zerlegung">Cholesky-Zerlegung</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=LL^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo>=</mo>
<mi>L</mi>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B=LL^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3eeed6240be366ba738fd96df792fff520355fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.417ex; height:2.676ex;" alt="{\displaystyle B=LL^{T}}" loading="lazy"></span> zerlegen. Dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{-1}A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B^{-1}A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/228101f54cf73292547a5fc308923edae1690248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.84ex; height:2.676ex;" alt="{\displaystyle B^{-1}A}" loading="lazy"></span> <a href="%C3%84hnlichkeit_(Matrix)" title="Ähnlichkeit (Matrix)">ähnlich</a> zu einer Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:=L^{-1}A(L^{-1})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>:=</mo>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H:=L^{-1}A(L^{-1})^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad70284af321274145b083048dae8ba529c01435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.582ex; height:3.176ex;" alt="{\displaystyle H:=L^{-1}A(L^{-1})^{T}}" loading="lazy"></span>. Die Inverse von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> lässt sich sehr effizient berechnen, da <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> eine Dreiecksmatrix ist. Bestimmt man nun die Eigenwerte von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"></span>, so sind dies auch die Eigenwerte von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{-1}A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B^{-1}A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/228101f54cf73292547a5fc308923edae1690248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.84ex; height:2.676ex;" alt="{\displaystyle B^{-1}A}" loading="lazy"></span>.
</p><p>Für beliebige Matrizen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"></span> kann auch der <a href="QZ-Algorithmus" title="QZ-Algorithmus">QZ-Algorithmus</a> genutzt werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Betrachte das verallgemeinerte Eigenwertproblem
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}4&4&8\\4&0&4\\8&4&4\end{bmatrix}}x=\lambda {\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix}}x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}4&4&8\\4&0&4\\8&4&4\end{bmatrix}}x=\lambda {\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix}}x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ce5718609f0cd2574230d08c83d74bd51cb2739.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:36.507ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}4&4&8\\4&0&4\\8&4&4\end{bmatrix}}x=\lambda {\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix}}x}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Naiver_Ansatz">Naiver Ansatz</h3></div>
<p>Die Berechnung der Inversen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> ergibt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{-1}={\frac {1}{4}}{\begin{bmatrix}3&2&1\\2&4&2\\1&2&3\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B^{-1}={\frac {1}{4}}{\begin{bmatrix}3&2&1\\2&4&2\\1&2&3\end{bmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9af0c657a3be7341a5c149a295b100e953046e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.179ex; height:9.176ex;" alt="{\displaystyle B^{-1}={\frac {1}{4}}{\begin{bmatrix}3&2&1\\2&4&2\\1&2&3\end{bmatrix}}}" loading="lazy"></span></dd></dl>
<p>und damit
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{-1}A={\begin{bmatrix}7&4&9\\10&4&10\\9&4&7\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>10</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>10</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B^{-1}A={\begin{bmatrix}7&4&9\\10&4&10\\9&4&7\end{bmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f44c588b1447cfbf23ee3093f6db099dc1f5d6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.248ex; height:9.176ex;" alt="{\displaystyle B^{-1}A={\begin{bmatrix}7&4&9\\10&4&10\\9&4&7\end{bmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Die Eigenwerte dieser Matrix sind 20,7703 sowie -2 und - 0,7703.
</p>
<div class="mw-heading mw-heading3"><h3 id="Mittels_der_Cholesky-Zerlegung">Mittels der Cholesky-Zerlegung</h3></div>
<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"></span> sind symmetrisch und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> außerdem positiv definit. Die Cholesky-Zerlegung liefert die Matrix
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{T}={\begin{bmatrix}1{,}4142&-0{,}7071&0\\0&1{,}2247&-0{,}8165\\0&0&1{,}1547\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1,414</mn>
<mn>2</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>0,707</mn>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1,224</mn>
<mn>7</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>0,816</mn>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1,154</mn>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L^{T}={\begin{bmatrix}1{,}4142&-0{,}7071&0\\0&1{,}2247&-0{,}8165\\0&0&1{,}1547\end{bmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdaf40a5385f9e84471670d34c6f01b738ae39f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.562ex; height:9.176ex;" alt="{\displaystyle L^{T}={\begin{bmatrix}1{,}4142&-0{,}7071&0\\0&1{,}2247&-0{,}8165\\0&0&1{,}1547\end{bmatrix}}}" loading="lazy"></span>.
</p><p>Dann ist
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:=L^{-1}A(L^{-1})^{T}={\begin{bmatrix}2{,}0000&3{,}4641&7{,}3485\\3{,}4641&3{,}3333&8{,}0139\\7{,}3485&8{,}0139&12{,}6667\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>:=</mo>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>A</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2,000</mn>
<mn>0</mn>
</mtd>
<mtd>
<mn>3,464</mn>
<mn>1</mn>
</mtd>
<mtd>
<mn>7,348</mn>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3,464</mn>
<mn>1</mn>
</mtd>
<mtd>
<mn>3,333</mn>
<mn>3</mn>
</mtd>
<mtd>
<mn>8,013</mn>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7,348</mn>
<mn>5</mn>
</mtd>
<mtd>
<mn>8,013</mn>
<mn>9</mn>
</mtd>
<mtd>
<mn>12,666</mn>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H:=L^{-1}A(L^{-1})^{T}={\begin{bmatrix}2{,}0000&3{,}4641&7{,}3485\\3{,}4641&3{,}3333&8{,}0139\\7{,}3485&8{,}0139&12{,}6667\end{bmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/841a0eead4e8f1ba5d74f85fb122dc2585cf4eac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:50.717ex; height:9.176ex;" alt="{\displaystyle H:=L^{-1}A(L^{-1})^{T}={\begin{bmatrix}2{,}0000&3{,}4641&7{,}3485\\3{,}4641&3{,}3333&8{,}0139\\7{,}3485&8{,}0139&12{,}6667\end{bmatrix}}}" loading="lazy"></span>.
</p><p>Die Eigenwerte dieser Matrix sind wie zu erwarten mit den oben berechneten Eigenwerten identisch.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Peter_Knabner" title="Peter Knabner">Peter Knabner</a>, <a href="Wolf_Barth_(Mathematiker)" title="Wolf Barth (Mathematiker)">Wolf Barth</a>: <cite style="font-style:italic">Lineare Algebra</cite>. Grundlagen und Anwendungen (= <cite style="font-style:italic">Springer-Lehrbuch</cite>). Springer, Berlin 2012, ISBN 978-3-642-32185-6.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Verallgemeinertes+Eigenwertproblem&rft.au=Peter+Knabner%2C+Wolf+Barth&rft.btitle=Lineare+Algebra&rft.date=2012&rft.genre=book&rft.isbn=9783642321856&rft.place=Berlin&rft.pub=Springer&rft.series=Springer-Lehrbuch" style="display:none"> </span></li>
<li>Josef Stoer, <a href="Roland_Bulirsch" title="Roland Bulirsch">Roland Bulirsch</a>: <i>Numerische Mathematik 2.</i> 5. Auflage, Springer, Berlin/Heidelberg/New York 2005, ISBN 978-3-540-23777-8.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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