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Display information for equation id:math.2035.49 on revision:2035
* Page found: Stabilität und Langzeitverhalten (eq math.2035.49)
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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 class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>ϖ</mi><mo>¯</mo></mover></mrow></mrow><msup><mo>*</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>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>ω</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>:</mi></mtd></mtr><mtr><mtd></mtd><mtd><mn>0</mn><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>λ</mi><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo>−</mo><mi>λ</mi></mtd><mtd><mo>−</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>−</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>−</mo><mi>λ</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo>=</mo><mo>−</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>+</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></mtd><mtd><mo>⇒</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>=</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>/</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>=</mo><mo>±</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></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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