LineareAlgebra.Matrizen01 History

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September 17, 2014, at 11:47 PM by 217.250.78.165 -
Changed lines 23-24 from:

{$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $}

to:

\[ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) \]

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September 17, 2014, at 06:10 PM by 80.140.80.73 -
Changed lines 13-14 from:
\vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right)
to:

\[ \vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right) \]

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A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right)
to:

\[ A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right) \]

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{$$ \vec{k}_1=M\cdot\vec{k}_0 =

to:

\[ \vec{k}_1=M\cdot\vec{k}_0 =

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\left(\begin{array}{c}27600 \\10800\\21600\end{array} \right) $$}

to:

\left(\begin{array}{c}27600 \\10800\\21600\end{array} \right) \]

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September 17, 2014, at 06:08 PM by 80.140.80.73 -
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M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
to:

\[ M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \]

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September 17, 2014, at 06:04 PM by 80.140.80.73 -
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Definition: Ein rechteckiges Zahlenschema mit n Zeilen und m Spalten nennt man (n,m)-Matrix oder n\times m-Matrix (n "Kreuz" m).

to:

Definition: Ein rechteckiges Zahlenschema mit n Zeilen und m Spalten nennt man $(n,m)$-Matrix oder $n\times m$-Matrix (n "Kreuz" m).

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September 17, 2014, at 05:46 PM by 80.140.80.73 -
Changed lines 13-14 from:

\vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right)

to:
\vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right)
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A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right)

to:
A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right)
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September 17, 2014, at 05:45 PM by 80.140.80.73 -
Changed line 7 from:

M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)

to:
M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
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September 17, 2014, at 05:44 PM by 80.140.80.73 -
Changed lines 13-14 from:
\vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right)
to:

\vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right)

Changed lines 16-17 from:
A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right)
to:

A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right)

Changed lines 23-24 from:

{$$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $$}

to:

{$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $}

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September 17, 2014, at 05:43 PM by 80.140.80.73 -
Changed line 7 from:
M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
to:

M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)

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September 17, 2014, at 05:43 PM by 80.140.80.73 -
Changed lines 7-8 from:

\[ M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \]

to:
M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
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\[ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) \]

to:

{$$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $$}

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September 17, 2014, at 05:41 PM by 80.140.80.73 -
Changed line 7 from:
M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
to:

\[ M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \]

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September 17, 2014, at 05:40 PM by 80.140.80.73 -
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September 17, 2014, at 05:37 PM by 80.140.80.73 -
Changed lines 23-24 from:

{$$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $$}

to:

\[ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) \]

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September 26, 2013, at 10:08 PM by 217.250.82.60 -
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a) In unserem Prozess mit der Käuferwanderung erhielten wir eine Übergangsmatrix M mit den Übergangsraten und einen Startvektor \vec{k}_0 mit den Käuferzahlen zu Beginn:

to:

a) In unserem Prozess mit der Käuferwanderung erhielten wir eine Übergangsmatrix M mit den Übergangsraten und einen Startvektor \vec{k}_0 mit den Käuferzahlen zu Beginn:

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September 26, 2013, at 10:08 PM by 217.250.82.60 -
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{$$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $$}

to:

{$$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $$}

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{$$ \vec{k}_1=M\cdot\vec{k}_0 =

to:

{$$ \vec{k}_1=M\cdot\vec{k}_0 =

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\left(\begin{array}{c}27600 \\10800\\21600\end{array} \right) $$}

to:

\left(\begin{array}{c}27600 \\10800\\21600\end{array} \right) $$}

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September 26, 2013, at 10:06 PM by 217.250.82.60 -
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$$

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{$$

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$$

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$$}

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$$

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{$$

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$$

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$$}

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September 26, 2013, at 08:34 PM by 217.250.73.73 -
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$$ M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) $$

to:
M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
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December 02, 2009, at 08:05 PM by 84.173.91.145 -
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M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)
to:

$$ M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) $$

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November 15, 2009, at 07:57 PM by Rutz -
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November 14, 2009, at 12:30 PM by 84.173.117.20 -
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Matrizen02

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Matrizen01 >>

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Matrizen01 >>

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November 14, 2009, at 09:27 AM by 84.173.86.193 -
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b) \left(\begin{array}{ccc}2 & 5 & -3 \\ -1 &2&7\end{array}\right)\cdot \left(\begin{array}{c}12\\ -6 \\ 8\end{array}\right) = \left(\begin{array}{ccccc} 2\cdot 12& +& 5\cdot(-6)&+&(-3)\cdot8\\(-1)\cdot12&+&2\cdot(-6)&+&7\cdot 8\end{array}\right) = \left(\begin{array}{c}-30\\32\end{array}\right)

to:

b) \left(\begin{array}{ccc}2 & 5 & -3 \\ -1 &2&7\end{array}\right)\cdot \left(\begin{array}{c}12\\ -6 \\ 8\end{array}\right) = \left(\begin{array}{ccccc} 2\cdot 12& +& 5\cdot(-6)&+&(-3)\cdot8\\(-1)\cdot12&+&2\cdot(-6)&+&7\cdot 8\end{array}\right) = \left(\begin{array}{c}-30\\32\end{array}\right)

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November 13, 2009, at 10:19 PM by 84.173.91.254 -
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%right Matrizen02

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Matrizen02

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November 13, 2009, at 10:19 PM by 84.173.91.254 -
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%right Matrizen02

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November 13, 2009, at 09:04 PM by 84.173.91.254 -
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Matrix-Vektor-Multiplikation

Definition: Ein rechteckiges Zahlenschema mit n Zeilen und m Spalten nennt man (n,m)-Matrix oder n\times m-Matrix (n "Kreuz" m).

M=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right)

Für das Matrixelement in der i-ten Zeile und der j-ten Spalte schreibt man a_{ij}. Die Matrix wird meist mit einem Großbuchstaben bezeichnet und das Zahlenschema in (runde oder eckige) Klammern gesetzt.

Definition: Eine (n,1)-Matrix nennt man auch n-dimensionalen Vektor oder genauer Spaltenvektor. Eine (1,m)-Matrix heißt entsprechend Zeilenvektor. Wir verwenden für Vektoren kleine Buchstaben mit Pfeilen darüber.

\vec{v}=\left(\begin{array}{c}v_1 \\ \vdots \\ v_n \end{array}\right)

Definition: Das Produkt aus einer (n,m)-Matrix A und einem m-dimensionalen Vektor \vec{v} ist folgendermaßen erklärt:

A\cdot\vec{v}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1m}\\\vdots &&\vdots\\ a_{n1}&\cdots&a_{nm}\end{array}\right) \cdot \left(\begin{array}{c}v_1 \\ \vdots \\ v_m \end{array}\right)= \left(\begin{array}{c}a_{11}\cdot v_1 + a_{12}\cdot v_2 + \cdots + a_{1m}\cdot v_m\\ a_{21}\cdot v_1 + a_{22}\cdot v_2 + \cdots + a_{2m}\cdot v_m\\ \vdots \\ a_{n1}\cdot v_1+ a_{n2}\cdot v_2+\cdots +a_{nm}\cdot v_m \end{array}\right)

Das Matrix-Vektor-Produkt ist also selbst wieder ein Vektor, allerdings mit n Einträgen.

Beispiele: a) In unserem Prozess mit der Käuferwanderung erhielten wir eine Übergangsmatrix M mit den Übergangsraten und einen Startvektor \vec{k}_0 mit den Käuferzahlen zu Beginn: $$ M = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \qquad \vec{k}_0=\left(\begin{array}{c}24000 \\12000\\24000\end{array} \right) $$ Zu den Käuferzahlen der nächsten Woche kam man dann durch Multiplikation der Matrix mit dem Käufervektor: $$ \vec{k}_1=M\cdot\vec{k}_0 = \left( \begin{array}{ccc}0,8&0,3&0,2\\0,1&0,5&0,1\\0,1&0,2&0,7\end{array} \right) \cdot \left(\begin{array}{c}24000 \\12000\\24000\end{array} \right)= \left(\begin{array}{c}27600 \\10800\\21600\end{array} \right) $$ b) \left(\begin{array}{ccc}2 & 5 & -3 \\ -1 &2&7\end{array}\right)\cdot \left(\begin{array}{c}12\\ -6 \\ 8\end{array}\right) = \left(\begin{array}{ccccc} 2\cdot 12& +& 5\cdot(-6)&+&(-3)\cdot8\\(-1)\cdot12&+&2\cdot(-6)&+&7\cdot 8\end{array}\right) = \left(\begin{array}{c}-30\\32\end{array}\right)

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