03 Matrizenmultiplikation

Rückführung: Matrix $\cdot$ Spalte und Zeile $\cdot$ Matrix

Matrix $\cdot$ Spalte = Spalte

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$$A\cdot \vec{x}=A\cdot \left[\begin{array}{c}{x}_1\\ x_2\\ \dots \end{array}\right]=x_1\cdot \left[\begin{array}{c}\text{1. Spalte}\\ \dots \\ \dots \end{array}\right]+x_2\cdot \left[\begin{array}{c}\text{2. Spalte}\\ \dots \\ \dots \end{array}\right]+\dots$$

Beispiel

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$$\begin{array}{r}\left[\begin{array}{ccc}1& 2& 1\\ 3& 8& 1\\ 0& 4& 1\end{array}\right]\cdot \left[\begin{array}{c}1\\ 2\\ 1\end{array}\right]=1\cdot \left[\begin{array}{c}1\\ 3\\ 0\end{array}\right]+2\cdot \left[\begin{array}{c}2\\ 8\\ 4\end{array}\right]+1\cdot \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}1+4+1\\ 3+16+1\\ 0+8+1\end{array}\right]=\left[\begin{array}{c}6\\ 20\\ 9\end{array}\right]\end{array}$$

Zeile $\cdot$ Matrix = Zeile

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$$\vec{x}\cdot A=\left[\begin{array}{ccc}{x}_1& x_2& \dots \end{array}\right]\cdot A=x_1\cdot \left[\begin{array}{c}\text{1. Zeile}\dots \end{array}\right]+x_2\cdot \left[\begin{array}{c}\text{2. Zeile}\dots \end{array}\right]+\dots$$

Beispiel

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$$\begin{array}{rl}& \left[\begin{array}{ccc}1& 2& 1\end{array}\right]\cdot \left[\begin{array}{ccc}1& 2& 1\\ 3& 8& 1\\ 0& 4& 1\end{array}\right]=1\cdot \left[\begin{array}{ccc}1& 2& 1\end{array}\right]+2\cdot \left[\begin{array}{ccc}3& 8& 1\end{array}\right]+1\cdot \left[\begin{array}{ccc}0& 4& 1\end{array}\right]\\ & \\ =& \left[\begin{array}{ccc}1& 2& 1\end{array}\right]+\left[\begin{array}{ccc}6& 16& 2\end{array}\right]+\left[\begin{array}{ccc}0& 4& 1\end{array}\right]\\ & \\ & \\ =& \left[\begin{array}{ccc}7& 22& 4\end{array}\right]\end{array}$$

Wichtige Bemerkungen

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• Bei der Matrizenmultiplikation darf die Reihenfolge nicht vertauscht werden

$$\begin{array}{r}A\cdot B\ne B\cdot A\end{array}$$

• Die Reihenfolge in der man sie auswertet ist egal

$$\begin{array}{r}A\cdot B\cdot C=(A\cdot B)\cdot C=A\cdot (B\cdot C)\end{array}$$

• Die Anzahl der Spalten $A$ und Anzahl der Zeilen $B$ müssen übereinstimmen. Es gilt:

  • $m$ = Anzahl der Reihen in $A$
  • $n$ = Anzahl der Spalten in $A$ = Anzahl der Zeilen in $B$
  • $p$ = Anzahl der Spalten in B

• Die Schreibweise für Matrixdimensionen lautet $A\in {\mathbb{R}}^{m\times n}$

1. und 2.Form (Spalten- und Reihenmultiplikation)

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• Zusammenführung von Spalten- und Reihenmultiplikation zu Matrizenmultiplikation

$$\begin{array}{rl}A\cdot B=\left[\begin{array}{c}\leftarrow {\vec{a}}_1\to \\ \leftarrow {\vec{a}}_2\to \\ \dots \\ \leftarrow {\vec{a}}_n\to \end{array}\right]\cdot \left[\begin{array}{cccc}\uparrow & \uparrow & & \uparrow \\ {\vec{b}}_1& {\vec{b}}_2& \dots & {\vec{b}}_n\\ \downarrow & \downarrow & & \downarrow \end{array}\right]& =\left[\begin{array}{cccc}\uparrow & \uparrow & & \uparrow \\ A\cdot {\vec{b}}_1& A\cdot {\vec{b}}_2& \dots & A\cdot {\vec{b}}_n\\ \downarrow & \downarrow & & \downarrow \end{array}\right]\\ & \\ & \\ & =\left[\begin{array}{c}\leftarrow {\vec{a}}_1\cdot B\to \\ \leftarrow {\vec{a}}_2\cdot B\to \\ \dots \\ \leftarrow {\vec{a}}_n\cdot B\to \end{array}\right]\end{array}$$

Beispiele

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1. Form (Spalten-Bild)

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$$\begin{array}{r}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\cdot \left[\begin{array}{cc}-1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{c}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\cdot \left[\begin{array}{c}-1\\ 1\end{array}\right],\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\cdot \left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}\right]=\left[\begin{array}{c}-1\cdot \left[\begin{array}{c}1\\ 3\end{array}\right]+1\cdot \left[\begin{array}{c}2\\ 4\end{array}\right],0\cdot \left[\begin{array}{c}1\\ 3\end{array}\right]+1\cdot \left[\begin{array}{c}2\\ 4\end{array}\right]\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 1& 4\end{array}\right]\end{array}$$

• Die Spalten von $A$ wirken auf $B$

2. Form (Zeilen-Bild)

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$$\begin{array}{r}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\cdot \left[\begin{array}{cc}-1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{c}\left[\begin{array}{cc}1& 2\end{array}\right]\cdot \left[\begin{array}{cc}-1& 0\\ 1& 1\end{array}\right]\\ \left[\begin{array}{cc}3& 4\end{array}\right]\cdot \left[\begin{array}{cc}-1& 0\\ 1& 1\end{array}\right]\end{array}\right]=\left[\begin{array}{c}1\cdot \left[\begin{array}{cc}-1& 0\end{array}\right]+2\cdot \left[\begin{array}{cc}1& 1\end{array}\right]\\ 3\cdot \left[\begin{array}{cc}-1& 0\end{array}\right]+4\cdot \left[\begin{array}{cc}1& 1\end{array}\right]\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 1& 4\end{array}\right]\end{array}$$

• Die Zeilen von $A$ wirken auf $B$

3. Form (Allgemeine Matrizenmultiplikation)

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Reelles Skalarprodukt

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• Hat man eine Zeile oder Spalte $\vec{a}$ und $\vec{b}$ so kann man das reelle Skalarprodukt so definieren:

$$\begin{array}{r}<\vec{a},\vec{b}>=\sum _{i=1}^n{a}_i\cdot b_i(=<\vec{b},\vec{a}>)\end{array}$$ $$\begin{array}{r}\Rightarrow A\cdot B=\left[\begin{array}{c}\leftarrow \overrightarrow{a_1}\to \\ \leftarrow \overrightarrow{a_2}\to \\ ⋮\\ \leftarrow \overrightarrow{a_n}\to \end{array}\right]\cdot \left[\begin{array}{cccc}\uparrow & \uparrow & & \uparrow \\ \overrightarrow{b_1}& \overrightarrow{b_2}& ...& \overrightarrow{b_n}\\ \downarrow & \downarrow & & \downarrow \end{array}\right]=\overrightarrow{a_1}\cdot \overrightarrow{b_1}+\overrightarrow{a_2}\cdot \overrightarrow{b_2}+\cdots +\overrightarrow{a_n}\cdot \overrightarrow{b_n}\end{array}$$

Nutzung des reellen Skalarprodukts für die Multiplikation

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• Es gilt: $A\in {\mathbb{R}}^{m\times n}$ und $B\in {\mathbb{R}}^{n\times p}$

$$\begin{array}{rl}A\cdot B=\left[\begin{array}{c}\leftarrow \overrightarrow{a_1}\to \\ \leftarrow \overrightarrow{a_2}\to \\ ⋮\\ \leftarrow \overrightarrow{a_m}\to \end{array}\right]\cdot \left[\begin{array}{cccc}\uparrow & \uparrow & & \uparrow \\ \overrightarrow{b_1}& \overrightarrow{b_2}& ...& \overrightarrow{b_p}\\ \downarrow & \downarrow & & \downarrow \end{array}\right]& =\left[\begin{array}{cccc}<\overrightarrow{a_1},\overrightarrow{b_1}>& <\overrightarrow{a_1},\overrightarrow{b_2}>& \dots & <\overrightarrow{a_1},\overrightarrow{b_p}>\\ <\overrightarrow{a_2},\overrightarrow{b_1}>& <\overrightarrow{a_2},\overrightarrow{b_2}>& \dots & <\overrightarrow{a_2},\overrightarrow{b_p}>\\ ⋮& ⋮& \ddots & ⋮\\ <\overrightarrow{a_m},\overrightarrow{b_1}>& <\overrightarrow{a_m},\overrightarrow{b_2}>& \dots & <\overrightarrow{a_m},\overrightarrow{b_p}>\end{array}\right]\\ & \\ & \\ & =\left[\begin{array}{ccccccc}\overrightarrow{a_{11}}\cdot \overrightarrow{b_{11}}+\overrightarrow{a_{12}}\cdot \overrightarrow{b_{12}}+\cdots +\overrightarrow{a_{1n}}\cdot \overrightarrow{b_{1n}}& & \overrightarrow{a_{11}}\cdot \overrightarrow{b_{21}}+\overrightarrow{a_{12}}\cdot \overrightarrow{b_{22}}+\cdots +\overrightarrow{a_{1n}}\cdot \overrightarrow{b_{2n}}& & \dots & & \overrightarrow{a_{11}}\cdot \overrightarrow{b_{p1}}+\overrightarrow{a_{12}}\cdot \overrightarrow{b_{p2}}+\cdots +\overrightarrow{a_{1n}}\cdot \overrightarrow{b_{pn}}\\ \overrightarrow{a_{21}}\cdot \overrightarrow{b_{11}}+\overrightarrow{a_{22}}\cdot \overrightarrow{b_{12}}+\cdots +\overrightarrow{a_{2n}}\cdot \overrightarrow{b_{1n}}& & \overrightarrow{a_{21}}\cdot \overrightarrow{b_{21}}+\overrightarrow{a_{22}}\cdot \overrightarrow{b_{22}}+\cdots +\overrightarrow{a_{2n}}\cdot \overrightarrow{b_{2n}}& & \dots & & \overrightarrow{a_{21}}\cdot \overrightarrow{b_{p1}}+\overrightarrow{a_{22}}\cdot \overrightarrow{b_{p2}}+\cdots +\overrightarrow{a_{2n}}\cdot \overrightarrow{b_{pn}}\\ ⋮& & ⋮& & \ddots & & ⋮\\ \overrightarrow{a_{m1}}\cdot \overrightarrow{b_{11}}+\overrightarrow{a_{m2}}\cdot \overrightarrow{b_{12}}+\cdots +\overrightarrow{a_{mn}}\cdot \overrightarrow{b_{1n}}& & \overrightarrow{a_{m1}}\cdot \overrightarrow{b_{21}}+\overrightarrow{a_{m2}}\cdot \overrightarrow{b_{22}}+\cdots +\overrightarrow{a_{mn}}\cdot \overrightarrow{b_{2n}}& & \dots & & \overrightarrow{a_{m1}}\cdot \overrightarrow{b_{p1}}+\overrightarrow{a_{m2}}\cdot \overrightarrow{b_{p2}}+\cdots +\overrightarrow{a_{mn}}\cdot \overrightarrow{b_{pn}}\end{array}\right]\end{array}$$

$\Rightarrow$ Die Endnismatrix $C$ hat $m$ Zeilen und $p$ Spalten $\Rightarrow (m\times n)\cdot (n\times p)=(m\times p)$ oder $A\in {\mathbb{R}}^{m\times n}\wedge B\in {\mathbb{R}}^{n\times p}=(A\cdot B)\in {\mathbb{R}}^{m\times p}$

Beispiel

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geg.:

$$\begin{array}{r}A=\left[\begin{array}{ccc}2& 3& -1\\ 0& 4& 1\end{array}\right]\text{und}B=\left[\begin{array}{cc}1& -1\\ -1& 0\\ 0& 2\end{array}\right]\text{mit}A\in {\mathbb{R}}^{m\times n},B\in {\mathbb{R}}^{n\times p}\end{array}$$ $$\begin{array}{rl}A\cdot B=\left[\begin{array}{ccc}2& 3& -1\\ 0& 4& 1\end{array}\right]\cdot \left[\begin{array}{cc}1& -1\\ -1& 0\\ 0& 2\end{array}\right]& =\left[\begin{array}{cc}2\cdot 1+3\cdot (-1)+(-1)\cdot 0& 2\cdot (-1)+3\cdot 0+(-1)\cdot 2\\ 0\cdot 1+4\cdot (-1)+1\cdot 0& 0\cdot (-1)+4\cdot 0+1\cdot 2\end{array}\right]\\ & \\ & =\left[\begin{array}{cc}-1& -4\\ -4& 2\end{array}\right]\end{array}$$

4. Form (Allgemeine Matrizenmultiplikation vertauscht)

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• Statt Zeilen $\cdot$ Spalten machen wir Spalten $\cdot$ Zeilen $\Rightarrow$ Gegenteil des Skalarprodukts

$$\begin{array}{r}A\cdot B=\left[\begin{array}{cccc}\uparrow & \uparrow & & \uparrow \\ \overrightarrow{a_1}& \overrightarrow{a_2}& ...& \overrightarrow{a_n}\\ \downarrow & \downarrow & & \downarrow \end{array}\right]\cdot \left[\begin{array}{c}\leftarrow \overrightarrow{b_1}\to \\ \leftarrow \overrightarrow{b_2}\to \\ ⋮\\ \leftarrow \overrightarrow{b_n}\to \end{array}\right]\end{array}$$ $$\begin{array}{r}\overrightarrow{a_i}\cdot \overrightarrow{b_i}=\left[\begin{array}{c}{a}_{i,1}\\ a_{i,2}\\ ⋮\\ a_{i,m}\end{array}\right]\cdot \left[\begin{array}{cccc}{b}_{i,1}& b_{i,2}& \dots & b_{i,p}\end{array}\right]=\left[\begin{array}{cccc}{a}_{i,1}\cdot b_{i,1}& a_{i,1}\cdot b_{i,2}& \dots & a_{i,1}\cdot b_{i,p}\\ a_{i,2}\cdot b_{i,1}& a_{i,2}\cdot b_{i,2}& \dots & a_{i,2}\cdot b_{i,p}\\ ⋮& ⋮& \ddots & ⋮\\ a_{i,m}\cdot b_{i,1}& a_{i,m}\cdot b_{i,2}& \dots & a_{i,m}\cdot b_{i,p}\end{array}\right]\end{array}$$

$\Rightarrow A\cdot B=\overrightarrow{a_1}\cdot \overrightarrow{b_1}+\overrightarrow{a_2}+\overrightarrow{b_2}+\cdots +\overrightarrow{a_n}\cdot \overrightarrow{b_n}$

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