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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gramsche Determinante</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>Man kann in der <a href="Matrizenrechnung" class="mw-redirect" title="Matrizenrechnung">Matrizenrechnung</a> nur <a href="Determinante" title="Determinante">Determinanten</a> von <i>quadratischen</i> Matrizen als Maß für die Volumenänderung ihrer Abbildung definieren. Für nichtquadratische Matrizen gibt es <a href="Minor_(Mathematik)" class="mw-redirect" title="Minor (Mathematik)">Minoren</a> und <b>Gramsche Determinanten</b> (nach <a href="J%C3%B8rgen_Pedersen_Gram" title="Jørgen Pedersen Gram">Jørgen Pedersen Gram</a>), die Ähnliches leisten.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Für alle Matrizen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in K^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in K^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab303433d6aa889f1f94258f6afb6a04ffea388d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.617ex; height:2.343ex;" alt="{\displaystyle A\in K^{m\times n}}" loading="lazy"></span> 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 m\geq n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>≥<!-- ≥ --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\geq n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b02f25e62da7fe3162ac80446437cdc1c0fd341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\geq n}" loading="lazy"></span> nennt man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Gram} (A)=\det(A^{T}A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Gram</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Gram} (A)=\det(A^{T}A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4b24d36734d29d36e7c2a0e564903e2cb964aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.4ex; height:3.176ex;" alt="{\displaystyle \operatorname {Gram} (A)=\det(A^{T}A)}" loading="lazy"></span> die Gramsche Determinante. Es gilt: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Gram} (A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Gram</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Gram} (A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e34e3082f9f068b88ab2ab81ae9eaed153cad76d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.387ex; height:2.843ex;" alt="{\displaystyle \operatorname {Gram} (A)}" loading="lazy"></span> ist für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6419d3aa99701ca996737b17a5e1174d53e6c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }" loading="lazy"></span> nie negativ und genau dann <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rang} A<n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>rang</mi>
<mo><!-- --></mo>
<mi>A</mi>
<mo><</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {rang} A<n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f57c6cd0799dbbec1c4f37405dfc57c22e8aa80a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.152ex; height:2.509ex;" alt="{\displaystyle \operatorname {rang} A<n}" loading="lazy"></span>, also wenn die Spalten 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> <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear abhängig</a> sind. Man kann die Gramsche Determinante auch nach dem <a href="Satz_von_Binet-Cauchy" title="Satz von Binet-Cauchy">Satz von Binet-Cauchy</a> als Summe über das Quadrat aller maximalen Minoren schreiben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Gramsche_Matrix">Gramsche Matrix</h2></div>
<p>Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbb {R} ^{m\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in \mathbb {R} ^{m\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8025ab71ecd74b01e89f6918676203ad16cdc4a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.202ex; height:2.343ex;" alt="{\displaystyle A\in \mathbb {R} ^{m\times n}}" loading="lazy"></span> sind die Einträge 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^{T}A=:M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mo>=:</mo>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A^{T}A=:M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4c632b07b5ccc92f1b58dcc82a62151399dbf8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.063ex; height:2.676ex;" alt="{\displaystyle A^{T}A=:M}" loading="lazy"></span> die <a href="Standardskalarprodukt" title="Standardskalarprodukt">kanonischen Skalarprodukte</a> der Spalten 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>. Hierzu betrachtet man die folgende Verallgemeinerung:
</p><p>Sei auf einem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen <a href="Vektorraum" title="Vektorraum">K-Vektorraum</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 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/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"></span> mit der <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</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 (v_{1},..,v_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (v_{1},..,v_{n})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09a6b0cbdf324c4758eb6685ab42e1c5d42a955a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.473ex; height:2.843ex;" alt="{\displaystyle (v_{1},..,v_{n})}" loading="lazy"></span> eine <a href="Bilinearform" title="Bilinearform">Bilinearform</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 \langle \cdot ,\cdot \rangle \colon V\times V\to K,\quad (v,w)\mapsto \langle v,w\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>:<!-- : --></mo>
<mi>V</mi>
<mo>×<!-- × --></mo>
<mi>V</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>K</mi>
<mo>,</mo>
<mspace width="1em"></mspace>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>v</mi>
<mo>,</mo>
<mi>w</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle \colon V\times V\to K,\quad (v,w)\mapsto \langle v,w\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebc8276070ab77c99069edad416bee8bb26ec833.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.506ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle \colon V\times V\to K,\quad (v,w)\mapsto \langle v,w\rangle }" loading="lazy"></span> definiert. Dann nennt man die Matrix
</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 M:={\begin{pmatrix}\langle v_{1},v_{1}\rangle &\cdots &\langle v_{1},v_{n}\rangle \\\vdots &&\vdots \\\langle v_{n},v_{1}\rangle &\cdots &\langle v_{n},v_{n}\rangle \end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>:=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd></mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
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<mo>,</mo>
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<mo>)</mo>
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<annotation encoding="application/x-tex">{\displaystyle M:={\begin{pmatrix}\langle v_{1},v_{1}\rangle &\cdots &\langle v_{1},v_{n}\rangle \\\vdots &&\vdots \\\langle v_{n},v_{1}\rangle &\cdots &\langle v_{n},v_{n}\rangle \end{pmatrix}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c76b10261163ce1bb3f5a9765c8acf78959e9c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:33.281ex; height:10.843ex;" alt="{\displaystyle M:={\begin{pmatrix}\langle v_{1},v_{1}\rangle &\cdots &\langle v_{1},v_{n}\rangle \\\vdots &&\vdots \\\langle v_{n},v_{1}\rangle &\cdots &\langle v_{n},v_{n}\rangle \end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>die zur Bilinearform <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 \langle \cdot ,\cdot \rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"></span> gehörige Gramsche Matrix, bzw. darstellende Matrix der Bilinearform. Letzte wird durch die Einträge der Gram-Matrix vollständig festgelegt.
Die Bilinearform <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 \langle \cdot ,\cdot \rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"></span> ist genau dann ein <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a>, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a> und <a href="Positiv_definit" class="mw-redirect" title="Positiv definit">positiv definit</a> ist.
</p><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 \langle \cdot ,\cdot \rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"></span> ein Skalarprodukt, <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_{1},\dots ,v_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{1},\dots ,v_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e6ddc943290a5e10aa10a064fb4d6a745f0fde7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.706ex; height:2.009ex;" alt="{\displaystyle v_{1},\dots ,v_{n}}" loading="lazy"></span> eine beliebige Menge von Vektoren aus <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>
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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
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</math></span><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"></span>, so bezeichnet man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> als die Gram-Matrix 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 v_{1},\dots ,v_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{1},\dots ,v_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e6ddc943290a5e10aa10a064fb4d6a745f0fde7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.706ex; height:2.009ex;" alt="{\displaystyle v_{1},\dots ,v_{n}}" loading="lazy"></span>. Eine wichtige Anwendung in diesem Fall ist das Kriterium der linearen Unabhängigkeit: Die Vektoren sind genau dann linear unabhängig, wenn ihre Gramsche Determinante (Determinante der Gram-Matrix) nicht Null ist. Da die Gramsche Determinante in diesem Falle nichtnegativ ist, kann man aus ihr die Wurzel ziehen und durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Vol} (v_{1},\dotsc ,v_{n}):={\sqrt {\det(M)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Vol</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>v</mi>
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<mn>1</mn>
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</msub>
<mo>,</mo>
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<mo>,</mo>
<msub>
<mi>v</mi>
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<mi>n</mi>
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<mo>:=</mo>
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<msqrt>
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Vol} (v_{1},\dotsc ,v_{n}):={\sqrt {\det(M)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a425e0418986925bd48985b0d32436d4f5b57a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.618ex; height:4.843ex;" alt="{\displaystyle \operatorname {Vol} (v_{1},\dotsc ,v_{n}):={\sqrt {\det(M)}}}" loading="lazy"></span></dd></dl>
<p>das <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionale Volumen des durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},\dots ,v_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{1},\dots ,v_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e6ddc943290a5e10aa10a064fb4d6a745f0fde7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.706ex; height:2.009ex;" alt="{\displaystyle v_{1},\dots ,v_{n}}" loading="lazy"></span> aufgespannten <a href="Parallelepiped" title="Parallelepiped">Spates</a> erklären.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Spatprodukt#Doppeltes_Spatprodukt" title="Spatprodukt">Spatprodukt#Doppeltes Spatprodukt</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Gerd Fischer: <cite style="font-style:italic">Lineare Algebra: Eine Einführung für Studienanfänger</cite>. 13. Auflage. Vieweg, 2002, ISBN 3-528-97217-3.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gramsche+Determinante&rft.au=Gerd+Fischer&rft.btitle=Lineare+Algebra%3A+Eine+Einf%C3%BChrung+f%C3%BCr+Studienanf%C3%A4nger&rft.date=2002&rft.edition=13&rft.genre=book&rft.isbn=3528972173&rft.pub=Vieweg" style="display:none"> </span></li>
<li><a href="Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>: <cite style="font-style:italic">Lineare Algebra</cite>. 6. Auflage. Vieweg, 2009, ISBN 978-3-528-56508-4.<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:Gramsche+Determinante&rft.au=Albrecht+Beutelspacher&rft.btitle=Lineare+Algebra&rft.date=2009&rft.edition=6.&rft.genre=book&rft.isbn=9783528565084&rft.pub=Vieweg" style="display:none"> </span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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