trigonalisierbarkeit

Eigenwerte, Eigenvektoren, Eigenraum

Diagonalisierung

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Definitheit prüfen

Integralgleichung

kettenregel

Lösbarkeit einer Matrix

Glechgewicht

Schiefe Ebene

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Flächenträgheit

Wechselstrom

Cauchy Verdichtungssatz 

Schaltvorgänge

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Spann

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spannung

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Basis bestimmen

Fourier Koeffizienten bestimmen

Reimann-Summen

Ungleichungen 

Statik in der Ebene

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Funktionsfolgen

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Funktionen bestimmen

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Gruppen

lineare abbildung 

quadratische ergänzung

Baustatik 1

Anfangswertprobleme :)

Differentialgleichungen

basis 

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Cholesky-Zerlegung

Bild & Urbild

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Transistor als Thermometer

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Epsilon Delta Kriterium

prinzip der virtuellen verrückung 

Linear Kombination

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Arbeit 

Cholesky Zerlegung

Polynomfunktionen beschreiben

Zu Maß-und Integrationstheorie: Majorisierte Konvergenz

Teilchenphysik und Festkörperphysik

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test

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Regel von de L'Hopital

Mathe

KRAFT

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Folgenkriterium

ϵ-δ Kriterium 

Beweis Stetigkeit 

Wahrheitstabelle

2x2 Matrix

Matrix invertieren

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Lineare Abbildunge 

Vorderrad fahradgabel lagerkräfte berechnen 

Spannungsregelung

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Test

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Jordan-Normalform

Ansatz der Störfunktion zur Bestimmung der Partikulären Lösung

Kurvenintegral

dlg mithilfe inhomogenen linearen system lösen

Inverse einer Matrix

Gleichmäßige Stetigkeit

addition

Regel von L´Hospital

Lineare Abbildungen

dimension eines Untervektorraumes bestimmen

Stationäre Stoffübertragung

Polplan, Prinzip der virtuellen Verrückungen

komplexe Ungleichungen 

Bidualräume

Dualräume

Taylorpolynome

Kern und Basis

Diagonalisieren einer Matrix

Rekursive Folgen

Exponentialreihe

Darstellungsmatrizen

restglied

Basiswechsel

Gamma Funktion

<p>Die Matrix ist nicht positiv definit.</p>

<p>Berechne die Hauptminoren und prüfe ihr Vorzeichen:</p>


<p>\begin{align*}<br />\operatorname{det}\left(a_{11}\right) &amp;=\operatorname{det}(-1) \\</p>
<p>&amp;=-1 &lt;0</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\operatorname{det}\left(\begin{array}{ll}<br />a_{11} &amp; a_{12} \\<br />a_{21} &amp; a_{22}<br />\end{array}\right)&amp;=\operatorname{det}\left(\begin{array}{cc}<br />-1 &amp; 1 \\<br />1 &amp; -1<br />\end{array}\right) \\</p>
<p>&amp;=0\end{align*}</p>


<p>\begin{align*}<br />\operatorname{det}(A) &amp;=\operatorname{det}\left(\begin{array}{ccc}<br />-1 &amp; 1 &amp; \sqrt{2} \\<br />1 &amp; -1 &amp; \sqrt{2} \\<br />\sqrt{2} &amp; \sqrt{2} &amp; 0<br />\end{array}\right) \\</p>
<p>&amp;=0+2+2-(-2-2+0) \\<br />&amp;=8 &gt;0<br />\end{align*}</p>


<p><mjx-container class="MathJax" jax="SVG"><svg alt="\begin{align*}
\operatorname{det}(A) &=\operatorname{det}\left(\begin{array}{ccc}
-1 & 1 & \sqrt{2} \\
1 & -1 & \sqrt{2} \\
\sqrt{2} & \sqrt{2} & 0
\end{array}\right) \\
&=0+2+2-(-2-2+0) \\
&=8 >0
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<p>Da nicht alle Hauptminoren positiv sind, ist die Matrix nicht positiv definit.</p>

<p>Berechne alle Hauptminoren und entscheide anschließend ob die Matrix positiv definit ist.</p>
<p> </p>
<p>$$A=\begin{align*}\left(\begin{array}{ccc}<br />-1 &amp; 1 &amp; \sqrt{2} \\<br />1 &amp; -1 &amp; \sqrt{2} \\<br />\sqrt{2} &amp; \sqrt{2} &amp; 0<br />\end{array}\right)\end{align*}$$</p>

<p>Berechne alle Hauptminoren und entscheide anschließend ob die Matrix positiv definit ist.</p>
<p> </p>
<p><mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="A=\begin{align*}\left(\begin{array}{ccc}
-1 & 1 & \sqrt{2} \\
1 & -1 & \sqrt{2} \\
\sqrt{2} & \sqrt{2} & 0
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Dies ist eine interaktive Aufgabe zu: Definitheit prüfen mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Eigenwerte - Definitheit prüfen | Aufgabe mit Lösung


<h2 class="content_sub_heading">Eigenwerten, Räume und Vektoren</h2>
<ol>
<li>Bestimme das charakteristische Polynom von $A$ über $\chi_{A}(\lambda)=\operatorname{det}(A-\lambda E_n)$. Ziehe dazu jeweils ein $\lambda$ von den Einträgen der Hauptdiagonalen ab und berechne anschließend die Determinante.</li>
<li>Die Eigenwerte entsprechen den Nullstellen des charakteristischen Polynoms.<br />$\chi_{A}(\lambda)=0 \quad \Rightarrow \quad \lambda_1, \ldots, \lambda_n$</li>
<li>Bestimme jeweils den Eigenraum zu jedem gefundenen Eigenwert mit $\operatorname{Eig}_{A}(\lambda_i)=\operatorname{ker}(A-\lambda_i \cdot E_n)$<br /><em>Du solltest hier bei jeder Berechnung unendlich viele Lösungen erhalten, also immer eine Lösungsmenge mit mindestens einem Parameter.</em></li>
<li>Einen passenden Eigenvektor erhälst du nun, indem du jeweils einen Vektor aus dem zugehörigen Eigenraum auswählst. Der Nullvektor ist dabei allerdings nicht zulässig.</li>
</ol>


Aufgabenstellung:

Lösungsweg:

Lösung: