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<title>Hesse-Matrix</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hesse-Matrix</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>Die nach <a href="Otto_Hesse" title="Otto Hesse">Otto Hesse</a> benannte <b>Hesse-Matrix</b> ist eine quadratische <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>, die in der <a href="Mehrdimensionale_Analysis" class="mw-redirect" title="Mehrdimensionale Analysis">mehrdimensionalen reellen Analysis</a> ein Analogon zur zweiten Ableitung einer Funktion ist.

</p><p>Die Hesse-Matrix taucht bei der <a href="Approximation" title="Approximation">Approximation</a> einer mehrdimensionalen Funktion in der <a href="Taylor-Entwicklung" class="mw-redirect" title="Taylor-Entwicklung">Taylor-Entwicklung</a> auf. Sie ist unter anderem in Zusammenhang mit der <a href="Optimierung_(Mathematik)" class="mw-redirect" title="Optimierung (Mathematik)">Optimierung</a> von Systemen von Bedeutung, die durch mehrere Parameter beschrieben werden, wie sie beispielsweise in den Wirtschaftswissenschaften, in der Physik, theoretischen Chemie oder in den Ingenieurwissenschaften häufig auftreten.

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

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<annotation encoding="application/x-tex">{\displaystyle f\colon D\subset \mathbb {R} ^{n}\to \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6471d5de8803e6389cbbd047669f06281a523fec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.524ex; height:2.676ex;" alt="{\displaystyle f\colon D\subset \mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span> eine <a href="Totale_Differenzierbarkeit" title="Totale Differenzierbarkeit">zweimal stetig differenzierbare Funktion</a>. Dann ist die Hesse-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 f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> am Punkt <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=(x_{1},\ldots ,x_{n})\in D}">

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<annotation encoding="application/x-tex">{\displaystyle x=(x_{1},\ldots ,x_{n})\in D}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cacdc46e2cb07269c11bdb0c632dc63eb8bf1bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.113ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},\ldots ,x_{n})\in D}" loading="lazy"></span> definiert durch

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{f}(x):=\left({\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}(x)\right)_{i,j=1,\dots ,n}={\begin{pmatrix}{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{1}}}(x)&{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}(x)&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}(x)\\[0.5em]{\frac {\partial ^{2}f}{\partial x_{2}\partial x_{1}}}(x)&{\frac {\partial ^{2}f}{\partial x_{2}\partial x_{2}}}(x)&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\partial x_{n}}}(x)\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}(x)&{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{2}}}(x)&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{n}}}(x)\end{pmatrix}}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04d705e48acb7356d6f3cf9dd39c135f342ed55a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:79.218ex; height:21.176ex;" alt="{\displaystyle \operatorname {H} _{f}(x):=\left({\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}(x)\right)_{i,j=1,\dots ,n}={\begin{pmatrix}{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{1}}}(x)&{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}(x)&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}(x)\\[0.5em]{\frac {\partial ^{2}f}{\partial x_{2}\partial x_{1}}}(x)&{\frac {\partial ^{2}f}{\partial x_{2}\partial x_{2}}}(x)&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\partial x_{n}}}(x)\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}(x)&{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{2}}}(x)&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{n}}}(x)\end{pmatrix}}.}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0150ba3c3d97277bee41eefe6a3c31fbe8aba897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.92ex; height:5.009ex;" alt="{\displaystyle {\tfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}}" loading="lazy"></span> werden die <a href="Partielle_Ableitung" title="Partielle Ableitung">zweiten partiellen Ableitungen</a> bezeichnet. Die Hesse-Matrix entspricht der <a href="Transponierte_Matrix" title="Transponierte Matrix">Transponierten</a> der <a href="Jacobi-Matrix" title="Jacobi-Matrix">Jacobi-Matrix</a> des <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a>, ist aber bei stetigen zweiten Ableitungen wegen der <a href="Satz_von_Schwarz" title="Satz von Schwarz">Vertauschbarkeit der Differentiationsreihenfolge</a> <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> so dass das Transponieren der Matrix keine Änderung bewirkt.

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

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<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/725b0e05a55a325c41bf1911f2fcd83dc7101102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:3.009ex;" alt="{\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=x^{3}+y^{3}-3xy}">

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)=x^{3}+y^{3}-3xy}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85c9bee95566e8e2f6ce8d64238c2d8ddefb3a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.633ex; height:3.176ex;" alt="{\displaystyle f(x,y)=x^{3}+y^{3}-3xy}" loading="lazy"></span> gilt</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial f}{\partial x}}(x,y)=3x^{2}-3y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd4fe30dfdc07040d2e59e28856f22b1340818b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.84ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\partial f}{\partial x}}(x,y)=3x^{2}-3y}" 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 {\tfrac {\partial f}{\partial y}}(x,y)=3y^{2}-3x}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial f}{\partial y}}(x,y)=3y^{2}-3x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/676da61d594a17c6fa3b1aa310a25fc3fd4cc2ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.809ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\partial f}{\partial y}}(x,y)=3y^{2}-3x}" loading="lazy"></span>,</dd>

<dd>und für die zweiten Ableitungen dementsprechend:</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial ^{2}f}{\partial x\partial x}}(x,y)=6x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179219afea8a1a1d174aee3162de91ca06dd93c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.5ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\partial ^{2}f}{\partial x\partial x}}(x,y)=6x}" loading="lazy"></span> und</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial ^{2}f}{\partial x\partial y}}(x,y)=-3}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39f3b4558db5b6c1d64355d439f377f2480612ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.855ex; height:4.843ex;" alt="{\displaystyle {\tfrac {\partial ^{2}f}{\partial x\partial y}}(x,y)=-3}" loading="lazy"></span>, beziehungsweise</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial ^{2}f}{\partial y\partial x}}(x,y)=-3}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial ^{2}f}{\partial y\partial y}}(x,y)=6y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/670a1320d60f26a0b4b2d81c0ac6e75ea5c93216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.079ex; height:4.843ex;" alt="{\displaystyle {\tfrac {\partial ^{2}f}{\partial y\partial y}}(x,y)=6y}" loading="lazy"></span>.</dd>

<dd>Somit ergibt sich die Hessematrix zu:</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{f}(x,y)={\begin{pmatrix}6x&-3\\-3&6y\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd1d7c9d5f4d89718cf4c8d84b8e10a968f8735e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.743ex; height:6.176ex;" alt="{\displaystyle \operatorname {H} _{f}(x,y)={\begin{pmatrix}6x&-3\\-3&6y\end{pmatrix}}}" loading="lazy"></span>.</dd></dl></dd></dl>

<ul><li>Die Funktion <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\colon \mathbb {R} ^{n}\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle r\colon \mathbb {R} ^{n}\to \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/475d3101582b3ea9856cc19fe484eb767cca8bd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.271ex; height:2.343ex;" alt="{\displaystyle r\colon \mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r(x)=\|x\|={\sqrt {\sum _{j=1}^{n}x_{j}^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle r(x)=\|x\|={\sqrt {\sum _{j=1}^{n}x_{j}^{2}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e226a307af78e2f1459ce6185f4f02393ac5b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.598ex; height:4.843ex;" alt="{\displaystyle \textstyle r(x)=\|x\|={\sqrt {\sum _{j=1}^{n}x_{j}^{2}}}}" loading="lazy"></span>, die jedem Vektor im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> seine <a href="Euklidische_Norm" title="Euklidische Norm">euklidische Norm</a> zuordnet, ist für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

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

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</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> zweimal stetig differenzierbar und es gilt nach der <a href="Kettenregel" title="Kettenregel">Kettenregel</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial r}{\partial x_{j}}}(x)={\frac {x_{j}}{\|x\|}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/342a924d3d485a980b71f323b106260a5548c8c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.122ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial r}{\partial x_{j}}}(x)={\frac {x_{j}}{\|x\|}}}" loading="lazy"></span></dd></dl></dd>

<dd>sowie weiter nach der <a href="Quotientenregel" title="Quotientenregel">Quotientenregel</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}r}{\partial x_{i}\partial x_{j}}}(x)={\frac {\delta _{ij}\|x\|-x_{j}{\frac {x_{i}}{\|x\|}}}{\|x\|^{2}}}={\frac {1}{\|x\|}}\delta _{ij}-{\frac {x_{i}x_{j}}{\|x\|^{3}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4151d5a71e134fbc65299d883225451f4b2887e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.063ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial ^{2}r}{\partial x_{i}\partial x_{j}}}(x)={\frac {\delta _{ij}\|x\|-x_{j}{\frac {x_{i}}{\|x\|}}}{\|x\|^{2}}}={\frac {1}{\|x\|}}\delta _{ij}-{\frac {x_{i}x_{j}}{\|x\|^{3}}}}" loading="lazy"></span>,</dd></dl></dd>

<dd>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 \delta _{ij}={\frac {\partial x_{j}}{\partial x_{i}}}}">

<msub>

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<mfrac>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}={\frac {\partial x_{j}}{\partial x_{i}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f31f7c4a26f1e4c6f3a08c6a74cc6d7d49c1d0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.002ex; height:6.176ex;" alt="{\displaystyle \delta _{ij}={\frac {\partial x_{j}}{\partial x_{i}}}}" loading="lazy"></span> das <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a> bezeichnet. In Matrixschreibweise folgt also

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{r}(x)={\frac {1}{\|x\|}}E_{n}-{\frac {1}{\|x\|^{3}}}xx^{T}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5091731032ebbebd4a7a605893def2c26032ab86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.813ex; height:6.009ex;" alt="{\displaystyle \operatorname {H} _{r}(x)={\frac {1}{\|x\|}}E_{n}-{\frac {1}{\|x\|^{3}}}xx^{T}}" loading="lazy"></span></dd></dl></dd>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span>-<a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</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 E_{n}}">

<msub>

</mrow>

</msub>

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</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy"></span>.</dd></dl>

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

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

<p>Die <a href="Taylor-Entwicklung" class="mw-redirect" title="Taylor-Entwicklung">Taylor-Entwicklung</a> einer zweimal stetig differenzierbaren Funktion <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 f\colon D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon D\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229e3a78dd4be03159691c3d4c84a9095784d26e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.529ex; height:2.509ex;" alt="{\displaystyle f\colon D\to \mathbb {R} }" 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 D\subseteq \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b709c793b75dcc64fb80372ca2bb82681a07452b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.919ex; height:2.509ex;" alt="{\displaystyle D\subseteq \mathbb {R} ^{n}}" loading="lazy"></span> um eine Entwicklungsstelle <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 D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/227a100dcc33dc54f1033c9e41f1b1a09dc31260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.995ex; height:2.176ex;" alt="{\displaystyle a\in D}" loading="lazy"></span> beginnt mit

</p>

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<annotation encoding="application/x-tex">{\displaystyle T(x)=f(a)+(x-a)^{T}\operatorname {grad} f(a)+{\frac {1}{2}}(x-a)^{T}\operatorname {H} _{f}(a)(x-a)+\ldots }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73f38f1e9aa6a07529b1b56d9a1f8f73fbf83c6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:65.767ex; height:5.176ex;" alt="{\displaystyle T(x)=f(a)+(x-a)^{T}\operatorname {grad} f(a)+{\frac {1}{2}}(x-a)^{T}\operatorname {H} _{f}(a)(x-a)+\ldots }" loading="lazy"></span></dd></dl>

<p>Die Terme zweiter Ordnung dieser Entwicklung sind also durch die <a href="Quadratische_Form" title="Quadratische Form">quadratische Form</a> gegeben, deren Matrix die an der Entwicklungsstelle ausgewertete Hesse-Matrix ist.

</p>

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

<p>Mit Hilfe der Hesse-Matrix lässt sich der Charakter der <a href="Kritischer_Punkt_(Mathematik)" title="Kritischer Punkt (Mathematik)">kritischen Punkte</a> einer reellen Funktion auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> bestimmen. Dazu bestimmt man für die zuvor ermittelten kritischen Punkte die <a href="Definitheit" title="Definitheit">Definitheit</a> der Hesse-Matrix.

</p>

<ul><li>Ist die Matrix an einer Stelle positiv definit, so befindet sich an diesem Punkt ein lokales <a href="Extremwert" title="Extremwert">Minimum</a> der Funktion.</li>

<li>Ist die Hesse-Matrix dort negativ definit, so handelt es sich um ein lokales <a href="Extremwert" title="Extremwert">Maximum</a>.</li>

<li>Ist sie indefinit, dann handelt es sich um einen <a href="Sattelpunkt" title="Sattelpunkt">Sattelpunkt</a> der Funktion.</li></ul>

<p>Falls die Hesse-Matrix an der untersuchten Stelle nur <a href="Definitheit" title="Definitheit">semidefinit</a> ist, so versagt dieses Kriterium und der Charakter des kritischen Punktes muss auf anderem Wege ermittelt werden. Welcher dieser Fälle vorliegt, kann – wie unter <a href="Definitheit" title="Definitheit">Definitheit</a> beschrieben – zum Beispiel mit Hilfe der <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> der <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenwerte</a> der Matrix oder ihrer <a href="Minor_(Mathematik)" class="mw-redirect" title="Minor (Mathematik)">Hauptminoren</a> entschieden werden.

</p><p>Beispiel: Die Funktion <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 f(x,y)=x^{2}-y^{2}}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)=x^{2}-y^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df03ddc768b381e4aa89a79eab73b8018740fc65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.144ex; height:3.176ex;" alt="{\displaystyle f(x,y)=x^{2}-y^{2}}" loading="lazy"></span> hat 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 (0,0)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"></span> einen kritischen Punkt, aber

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

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</mtr>

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<annotation encoding="application/x-tex">{\displaystyle H(f)(0,0)={\begin{pmatrix}2&0\\0&-2\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9697c848e4f42c82bc094d7ec892970b71c2be68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.046ex; height:6.176ex;" alt="{\displaystyle H(f)(0,0)={\begin{pmatrix}2&0\\0&-2\end{pmatrix}}}" loading="lazy"></span>

ist weder positiv noch negativ definit und auch nicht semidefinit, sondern indefinit. Die Funktion hat in diesem Punkt kein Extremum, sondern einen Sattelpunkt, in dem sich zwei <a href="H%C3%B6henlinie" title="Höhenlinie">Höhenlinien</a> schneiden.

</p>

<div class="mw-heading mw-heading3"><h3 id="Konvexität"><span id="Konvexit.C3.A4t"></span>Konvexität</h3></div>

<p>Es besteht zudem ein Zusammenhang zwischen der positiven <a href="Definitheit" title="Definitheit">Definitheit</a> der Hesse-Matrix und der <a href="Konvexe_und_konkave_Funktionen" title="Konvexe und konkave Funktionen">Konvexität</a> einer zweimal stetig differenzierbaren Funktion <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 f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>, die auf einer offenen, konvexen Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> definiert ist: Eine solche Funktion ist genau dann konvex, wenn ihre Hesse-Matrix <i>überall 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 D}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

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

</p><p>Entsprechend gilt: Eine zweimal stetig differenzierbare Funktion <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 f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> ist auf ihrer konvexen <a href="Definitionsmenge" title="Definitionsmenge">Definitionsmenge</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 D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> genau dann konkav, wenn ihre Hesse-Matrix negativ semidefinit ist. Ist die Hessematrix sogar negativ definit auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

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

</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 f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> auf ihrer Definitionsmenge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> höchstens ein globales Minimum auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> strikt konkav, so besitzt <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 f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> höchstens ein globales Maximum. Jedes lokale Maximum ist zugleich ihr (einziges) globales Maximum.<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-heading3"><h3 id="Laplace-Operator">Laplace-Operator</h3></div>

<p>Der <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> einer zweimal stetig differenzierbaren Funktion <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 f\colon D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon D\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229e3a78dd4be03159691c3d4c84a9095784d26e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.529ex; height:2.509ex;" alt="{\displaystyle f\colon D\to \mathbb {R} }" 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 D\subseteq \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b709c793b75dcc64fb80372ca2bb82681a07452b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.919ex; height:2.509ex;" alt="{\displaystyle D\subseteq \mathbb {R} ^{n}}" loading="lazy"></span> ist gleich der <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> ihrer Hesse-Matrix und daher unabhängig von der Wahl der Koordinaten:

</p>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=\mathrm {Spur} ({H}_{f})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88978d203de6242596a7e72bd2ca0d5c69f93c02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.979ex; height:3.009ex;" alt="{\displaystyle \Delta f=\mathrm {Spur} ({H}_{f})}" loading="lazy"></span></dd></dl>

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

<ul><li><a href="Ger%C3%A4nderte_Hesse-Matrix" title="Geränderte Hesse-Matrix">Geränderte Hesse-Matrix</a></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Otto_Forster" title="Otto Forster">Otto Forster</a>: <cite style="font-style:italic">Analysis 2</cite>. Differentialrechnung im R<sup>n</sup>, gewöhnliche Differentialgleichungen. 8. Auflage. Vieweg+Teubner Verlag, Wiesbaden 2008, ISBN 978-3-8348-0575-1, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>78</span>.<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:Hesse-Matrix&rft.au=Otto+Forster&rft.btitle=Analysis+2&rft.date=2008&rft.edition=8.&rft.genre=book&rft.isbn=9783834805751&rft.pages=78&rft.place=Wiesbaden&rft.pub=Vieweg%2BTeubner+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131102045601/http://www.math.uni-hamburg.de/home/oberle/skripte/optimierung/optim03.pdf"><i>Konvexe Funktionen.</i></a> <span style="white-space:nowrap;">S. 16</span>, archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">

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</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.math.uni-hamburg.de%2Fhome%2Foberle%2Fskripte%2Foptimierung%2Foptim03.pdf">Original</a></span> (nicht mehr online verfügbar) am <span style="white-space:nowrap;">2. November 2013</span><span>;</span><span class="Abrufdatum"> abgerufen am 16. September 2012</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AHesse-Matrix&rft.title=Konvexe+Funktionen&rft.description=Konvexe+Funktionen&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20131102045601%2Fhttp%3A%2F%2Fwww.math.uni-hamburg.de%2Fhome%2Foberle%2Fskripte%2Foptimierung%2Foptim03.pdf&rft.source=http://www.math.uni-hamburg.de/home/oberle/skripte/optimierung/optim03.pdf"> </span></span>

</li>

</ol>

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

<ul><li><a rel="nofollow" class="external text" href="https://statmath.wu.ac.at/~leydold/MOK/HTML/node134.html">Weiteres zum Zusammenhang Konvexität – Hesse-Matrix</a></li></ul>

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

<ul><li><a href="Konrad_K%C3%B6nigsberger" title="Konrad Königsberger">Konrad Königsberger</a>: <i>Analysis.</i> Band 2. 3. überarbeitete Auflage. Springer-Verlag, Berlin u. a. 2000, ISBN 3-540-66902-7.</li></ul></div><div><div class="zim-footer">

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