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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Rayleigh-Quotient</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>Der <b>Rayleigh-Quotient</b>, auch <b>Rayleigh-Koeffizient</b> genannt, ist ein Objekt aus der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>, das nach dem Physiker <a href="John_William_Strutt%2C_3._Baron_Rayleigh" class="mw-redirect" title="John William Strutt, 3. Baron Rayleigh">John William Strutt, 3. Baron Rayleigh</a> benannt ist. Der Rayleigh-Quotient wird insbesondere zur <a href="Numerik" class="mw-redirect" title="Numerik">numerischen</a> Berechnung von <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerten</a> einer quadratischen <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> verwendet.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathbb {K} }^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in {\mathbb {K} }^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f4ffe5192354872404a4d1e7665a6ca967c1356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.875ex; height:2.343ex;" alt="{\displaystyle A\in {\mathbb {K} }^{n\times n}}" loading="lazy"></span> eine reelle <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische</a> oder komplexe <a href="Hermitesche_Matrix" title="Hermitesche Matrix">hermitesche Matrix</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 x\in {\mathbb {K} }^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\in {\mathbb {K} }^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9b5ca89d8a1c58b9ce3f1b69c67d486fb4b7686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.197ex; height:2.343ex;" alt="{\displaystyle x\in {\mathbb {K} }^{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> ein <a href="Vektor" title="Vektor">Vektor</a>, dann ist der Rayleigh-Quotient von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> zum Vektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> definiert durch<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</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 R_{A}(x)={\frac {x^{*}Ax}{x^{*}x}}\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mi>A</mi>
<mi>x</mi>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{A}(x)={\frac {x^{*}Ax}{x^{*}x}}\,.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/408f4ef1339a1179a11f3a1ca473816a2038451c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.793ex; height:5.343ex;" alt="{\displaystyle R_{A}(x)={\frac {x^{*}Ax}{x^{*}x}}\,.}" loading="lazy"></span></dd></dl>
<p>Hierbei bezeichnet <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^{*}={\overline {x}}^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{*}={\overline {x}}^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93fecfcee8b82f927cb7512ff115d0782e6ede11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle x^{*}={\overline {x}}^{T}}" loading="lazy"></span> den <a href="Adjungierte_Matrix" title="Adjungierte Matrix">adjungierten</a> Vektor 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 x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>. Der <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Bildbereich</a> des Rayleigh-Quotienten ist genau der <a href="Numerischer_Wertebereich_(Hilbertraum)" title="Numerischer Wertebereich (Hilbertraum)">numerische Wertebereich</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<p>Bei einer <a href="Skalarmultiplikation" title="Skalarmultiplikation">Multiplikation</a> des Vektors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> mit einem Skalar <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha \neq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5770f8ec95300cbfde18eb59c49a11f12adbcd91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.749ex; height:2.676ex;" alt="{\displaystyle \alpha \neq 0}" loading="lazy"></span> ändert sich der Rayleigh-Quotient nicht: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{A}(\alpha x)=R_{A}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{A}(\alpha x)=R_{A}(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39fb993eaaa34b381ad542b52c4d796921815d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.322ex; height:2.843ex;" alt="{\displaystyle R_{A}(\alpha x)=R_{A}(x)}" loading="lazy"></span>, er ist also eine <a href="Homogene_Funktion" title="Homogene Funktion">homogene Funktion</a> vom Grad 0.
</p><p>Der Rayleigh-Quotient hat eine enge Beziehung zu den <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerten</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>. 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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> ein <a href="Eigenvektor" class="mw-redirect" title="Eigenvektor">Eigenvektor</a> der Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> 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 \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> der zugehörige Eigenwert, dann gilt:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{A}(v)={\frac {v^{*}Av}{v^{*}v}}={\frac {v^{*}\lambda v}{v^{*}v}}=\lambda \,.}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle R_{A}(v)={\frac {v^{*}Av}{v^{*}v}}={\frac {v^{*}\lambda v}{v^{*}v}}=\lambda \,.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e6e7182831877ed9f752b49f43e2e630345c10c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.24ex; height:5.343ex;" alt="{\displaystyle R_{A}(v)={\frac {v^{*}Av}{v^{*}v}}={\frac {v^{*}\lambda v}{v^{*}v}}=\lambda \,.}" loading="lazy"></span></dd></dl>
<p>Durch den Rayleigh-Quotienten wird also jeder Eigenvektor von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> auf den dazugehörigen Eigenwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
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<mi>λ<!-- λ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> abgebildet. Diese Eigenschaft wird unter anderem in der numerischen Berechnung von Eigenwerten benutzt. Insbesondere gilt für eine <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische</a> oder <a href="Hermitesche_Matrix" title="Hermitesche Matrix">hermitesche</a> Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> mit dem kleinsten Eigenwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\rm {min}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>λ<!-- λ --></mi>
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<mi mathvariant="normal">i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \lambda _{\rm {min}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6abcdaaee44fb79033a413dc60db197d65091bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.328ex; height:2.509ex;" alt="{\displaystyle \lambda _{\rm {min}}}" loading="lazy"></span> und dem größten Eigenwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\rm {max}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="normal">m</mi>
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<mi mathvariant="normal">x</mi>
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<annotation encoding="application/x-tex">{\displaystyle \lambda _{\rm {max}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c0851115f0de94c4f13fb906802a3c96d4f2cf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.646ex; height:2.509ex;" alt="{\displaystyle \lambda _{\rm {max}}}" loading="lazy"></span> nach dem <a href="Satz_von_Courant-Fischer" title="Satz von Courant-Fischer">Satz von Courant-Fischer</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\rm {min}}\leq R_{A}(x)\leq \lambda _{\rm {max}}\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>λ<!-- λ --></mi>
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<mo>≤<!-- ≤ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \lambda _{\rm {min}}\leq R_{A}(x)\leq \lambda _{\rm {max}}\,.}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e39c727aa530209b0f42cb17efd85d8448b862c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.573ex; height:2.843ex;" alt="{\displaystyle \lambda _{\rm {min}}\leq R_{A}(x)\leq \lambda _{\rm {max}}\,.}" loading="lazy"></span></dd></dl>
<p>Die Berechnung des kleinsten bzw. größten Eigenwerts ist damit äquivalent zum Auffinden des Minimums bzw. Maximums des Rayleigh-Quotienten. Das lässt sich unter geeigneten Voraussetzungen auch noch auf den unendlichdimensionalen Fall verallgemeinern und ist als <a href="Rayleigh-Ritz-Prinzip" title="Rayleigh-Ritz-Prinzip">Rayleigh-Ritz-Prinzip</a> bekannt.
</p><p>Die Eigenvektoren hermitescher Matrizen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> bilden die <a href="Station%C3%A4rer_Punkt" class="mw-redirect" title="Stationärer Punkt">stationären Punkte</a> des Rayleigh-Quotienten. Dies gilt nicht für asymmetrische Matrizen.
Deswegen führte <a href="Alexander_Markowitsch_Ostrowski" title="Alexander Markowitsch Ostrowski">Ostrowski</a> 1958/59 den sogenannten 2-seitigen Rayleigh-Quotienten
</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 R_{A}(x,y)={\frac {y^{*}Ax}{y^{*}x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>R</mi>
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<mi>A</mi>
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<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle R_{A}(x,y)={\frac {y^{*}Ax}{y^{*}x}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cc2248d2e9b3a2a3af4bc09082d5defb247064a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.779ex; height:6.009ex;" alt="{\displaystyle R_{A}(x,y)={\frac {y^{*}Ax}{y^{*}x}}}" loading="lazy"></span></dd></dl>
<p>ein, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{*}x\neq 0}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle y^{*}x\neq 0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35827c9d31344c66737777d866fc5c48d23618ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.805ex; height:2.843ex;" alt="{\displaystyle y^{*}x\neq 0}" loading="lazy"></span>, der wiederum stationär an den Rechts- und <a href="Linkseigenvektor" class="mw-redirect" title="Linkseigenvektor">Linkseigenvektoren</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> 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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><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"></span> ist. Da für <a href="Normale_Matrix" title="Normale Matrix">normale Matrizen</a> Rechts- und Linkseigenvektoren übereinstimmen, fällt der 2-seitige mit dem (einseitigen) Rayleigh-Quotienten in diesem Fall zusammen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwendung_in_der_Numerischen_Mathematik">Verwendung in der Numerischen Mathematik</h2></div>
<p>Bei numerischen Verfahren zur Lösung von Eigenwertproblemen, die, wie beispielsweise die <a href="Vektoriteration" class="mw-redirect" title="Vektoriteration">Vektoriteration</a> oder die <a href="Inverse_Iteration" title="Inverse Iteration">inverse Iteration</a>, primär Eigenvektornäherungen berechnen, lassen sich mit Hilfe des Rayleigh-Quotienten zusätzlich auch Eigenwertnäherungen bestimmen. Bei der inversen Iteration wird ein Parameter <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 \theta }">
<semantics>
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<mi>θ<!-- θ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span>, der <i>Shift</i>, benötigt. Wird <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 \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>θ<!-- θ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> in jedem Iterationsschritt als Rayleigh-Quotient der aktuellen Eigenvektornäherung gewählt, ergibt dies das sogenannte Rayleigh-Quotienten-Verfahren.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Gisela Engeln-Müllges, Klaus Niederdrenk, Reinhard Wodicka: <cite style="font-style:italic">Numerik-Algorithmen</cite>. 10. Auflage. Springer-Verlag, Berlin, Heidelberg 2011, ISBN 978-3-642-13472-2, Abschnitt 7.4 Konvergenzverbesserung mit Hilfe des Rayleigh-Quotienten im Falle hermitescher Matrizen.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&amp;rfr_id=info:sid/de.wikipedia.org:Rayleigh-Quotient&amp;rft.atitle=Abschnitt+7.4+Konvergenzverbesserung+mit+Hilfe+des+Rayleigh-Quotienten+im+Falle+hermitescher+Matrizen&amp;rft.au=Gisela+Engeln-M%C3%BCllges%2C+Klaus+Niederdrenk%2C+Reinhard+Wodicka&amp;rft.btitle=Numerik-Algorithmen&amp;rft.date=2011&amp;rft.edition=10&amp;rft.genre=bookitem&amp;rft.isbn=9783642134722&amp;rft.place=Berlin%2C+Heidelberg&amp;rft.pub=Springer-Verlag" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Lloyd_Nicholas_Trefethen" title="Lloyd Nicholas Trefethen">Lloyd N. Trefethen</a>, David Bau: <i>Numerical Linear Algebra.</i> SIAM, 1997, ISBN 978-0-89871-361-9, S. 207–210 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=bj-Lu6zjWbEC&amp;pg=PA207#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).</span>
</li>
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