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Display information for equation id:math.1357.118 on revision:1357
* Page found: Dynamische Systeme und deterministisches Chaos (eq math.1357.118)
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TeX (original user input):
\begin{align}
& \bar{\varpi }{{*}^{(3)}}=\left( \begin{matrix}
0 & 0 & \omega \\
\end{matrix} \right): \\
& 0=\det (A-\lambda 1)=\left| \begin{matrix}
-\lambda & -{{k}_{1}}\omega & 0 \\
{{k}_{2}}\omega & -\lambda & {{0}_{{}}} \\
0 & 0 & -\lambda \\
\end{matrix} \right|=-\lambda \left( {{\lambda }^{2}}+{{k}_{1}}{{k}_{2}}{{\omega }^{2}} \right) \\
& \Rightarrow {{\lambda }_{1}}^{(3)}=0,{{\lambda }_{2/3}}^{(2)}=\pm i\omega \sqrt{{{k}_{1}}{{k}_{2}}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{\bar {\varpi }}{{*}^{(3)}}=\left({\begin{matrix}0&0&\omega \\\end{matrix}}\right):\\&0=\det(A-\lambda 1)=\left|{\begin{matrix}-\lambda &-{{k}_{1}}\omega &0\\{{k}_{2}}\omega &-\lambda &{{0}_{}}\\0&0&-\lambda \\\end{matrix}}\right|=-\lambda \left({{\lambda }^{2}}+{{k}_{1}}{{k}_{2}}{{\omega }^{2}}\right)\\&\Rightarrow {{\lambda }_{1}}^{(3)}=0,{{\lambda }_{2/3}}^{(2)}=\pm i\omega {\sqrt {{{k}_{1}}{{k}_{2}}}}\\\end{aligned}}
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<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mover><mi>ϖ</mi><mo>¯</mo></mover><msup><mo stretchy="false" lspace="0" rspace="0">∗</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN"></mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>ω</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">:</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mn>0</mn><mo stretchy="false">=</mo><mo>det</mo><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">−</mo><mi>λ</mi><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN"></mo><mtable><mtr><mtd><mo stretchy="false">−</mo><mi>λ</mi></mtd><mtd><mo stretchy="false">−</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mi>ω</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mi>ω</mi></mtd><mtd><mo stretchy="false">−</mo><mi>λ</mi></mtd><mtd><msub><mn>0</mn><mrow data-mjx-texclass="ORD"></mrow></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo stretchy="false">−</mo><mi>λ</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo stretchy="false">=</mo><mo stretchy="false">−</mo><mi>λ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>λ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msup><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mo stretchy="false">=</mo><mn>0</mn><mo>,</mo><msup><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mo lspace="0" rspace="0">/</mo><mn>3</mn></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mo stretchy="false">=</mo><mo stretchy="false">±</mo><mi>i</mi><mi>ω</mi><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mrow></msqrt></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>
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