Multiplikation von Matrizen

Wir betrachten den Fall, dass in einem linearen Gleichungssystem $\mathit{A}\mathit{y}=\mathit{d}(\mathit{i}=1,2,\dots ,\mathit{N})$ die Unbekannten ${\mathit{y}}_{\mathit{i}}$ gemäß eines zweiten linearen Gleichungssystems $\mathit{y}=\mathit{B}\mathit{x}$ von den Unbekannten ${\mathit{x}}_{\mathit{j}}(\mathit{j}=1,2,\dots ,\mathit{M})$ abhängig sind. Für den Fall $\mathit{N}=2$ und $\mathit{M}=3$ sieht das so aus:

$$\begin{array}{rcl}{\mathit{a}}_{11}\cdot {\mathit{y}}_1+{\mathit{a}}_{12}\cdot {\mathit{y}}_2& =& {\mathit{d}}_1\\ {\mathit{a}}_{21}\cdot {\mathit{y}}_1+{\mathit{a}}_{22}\cdot {\mathit{y}}_2& =& {\mathit{d}}_2\end{array}$$

und

$$\begin{array}{rcl}{\mathit{y}}_1& =& {\mathit{b}}_{11}\cdot {\mathit{x}}_1+{\mathit{b}}_{12}\cdot {\mathit{x}}_2+{\mathit{b}}_{13}\cdot {\mathit{x}}_3\\ {\mathit{y}}_2& =& {\mathit{b}}_{21}\cdot {\mathit{x}}_1+{\mathit{b}}_{22}\cdot {\mathit{x}}_2+{\mathit{b}}_{23}\cdot {\mathit{x}}_3\end{array}$$

Wir können beide Gleichungssysteme vereinigen, indem wir im oberen System ${\mathit{y}}_i$ entsprechend des unteren Systems substituieren. In der Matrixschreibweise hat das die Form $\mathit{A}\mathit{B}\mathit{x}=\mathit{d}$, was ausgeschrieben so aussieht:

$$\begin{array}{rcl}{\mathit{a}}_{11}\cdot \left({\mathit{b}}_{11}{\mathit{x}}_1+{\mathit{b}}_{12}\cdot {\mathit{x}}_2+{\mathit{b}}_{13}\cdot {\mathit{x}}_3\right)+{\mathit{a}}_{12}\cdot \left({\mathit{b}}_{21}\cdot {\mathit{x}}_1+{\mathit{b}}_{22}\cdot {\mathit{x}}_2+{\mathit{b}}_{23}\cdot {\mathit{x}}_3\right)& =& {\mathit{d}}_1\\ {\mathit{a}}_{21}\cdot \left({\mathit{b}}_{11}{\mathit{x}}_1+{\mathit{b}}_{12}\cdot {\mathit{x}}_2+{\mathit{b}}_{13}\cdot {\mathit{x}}_3\right)+{\mathit{a}}_{22}\cdot \left({\mathit{b}}_{21}\cdot {\mathit{x}}_1+{\mathit{b}}_{22}\cdot {\mathit{x}}_2+{\mathit{b}}_{23}\cdot {\mathit{x}}_3\right)& =& {\mathit{d}}_2\end{array}$$

Ausmultiplizieren und -sortieren nach den Unbekannten ${\mathit{x}}_j$ ergibt:

$$\begin{array}{rcl}\left({\mathit{a}}_{11}\cdot {\mathit{b}}_{11}+{\mathit{a}}_{12}\cdot {\mathit{b}}_{21}\right)\cdot {\mathit{x}}_1+\left({\mathit{a}}_{11}\cdot {\mathit{b}}_{12}+{\mathit{a}}_{12}\cdot {\mathit{b}}_{22}\right)\cdot {\mathit{x}}_2+\left({\mathit{a}}_{11}\cdot {\mathit{b}}_{13}+{\mathit{a}}_{12}\cdot {\mathit{b}}_{23}\right)\cdot {\mathit{x}}_3& =& {\mathit{d}}_1\\ \left({\mathit{a}}_{21}\cdot {\mathit{b}}_{11}+{\mathit{a}}_{22}\cdot {\mathit{b}}_{21}\right)\cdot {\mathit{x}}_1+\left({\mathit{a}}_{21}\cdot {\mathit{b}}_{12}+{\mathit{a}}_{22}\cdot {\mathit{b}}_{22}\right)\cdot {\mathit{x}}_2+\left({\mathit{a}}_{21}\cdot {\mathit{b}}_{13}+{\mathit{a}}_{22}\cdot {\mathit{b}}_{23}\right)\cdot {\mathit{x}}_3& =& {\mathit{d}}_2\end{array}$$

Die Summen in den Klammern definieren die Koeffizienten ${\mathit{c}}_{\mathit{i}\mathit{j}}$ eines dritten Gleichungssystems $\mathit{C}\mathit{x}=\mathit{d}$

$$\begin{array}{rcl}{\mathit{c}}_{11}\cdot {\mathit{x}}_1+{\mathit{c}}_{12}\cdot {\mathit{x}}_2+{\mathit{c}}_{13}\cdot {\mathit{x}}_3& =& {\mathit{d}}_1\\ {\mathit{c}}_{21}\cdot {\mathit{x}}_1+{\mathit{c}}_{22}\cdot {\mathit{x}}_2+{\mathit{c}}_{23}\cdot {\mathit{x}}_3& =& {\mathit{d}}_2,\end{array}$$

in dem für die Koeffizientenmatrix $\mathit{C}$ gilt:

$$\left(\begin{array}{ccc}{\mathit{c}}_{11}& {\mathit{c}}_{12}& {\mathit{c}}_{13}\\ {\mathit{c}}_{21}& {\mathit{c}}_{22}& {\mathit{c}}_{23}\end{array}\right)=\left(\begin{array}{ccc}{\mathit{a}}_{11}{\mathit{b}}_{11}+{\mathit{a}}_{12}{\mathit{b}}_{21}& {\mathit{a}}_{11}{\mathit{b}}_{12}+{\mathit{a}}_{12}{\mathit{b}}_{22}& {\mathit{a}}_{11}{\mathit{b}}_{13}+{\mathit{a}}_{12}{\mathit{b}}_{23}\\ {\mathit{a}}_{21}{\mathit{b}}_{11}+{\mathit{a}}_{22}{\mathit{b}}_{21}& {\mathit{a}}_{21}{\mathit{b}}_{12}+{\mathit{a}}_{22}{\mathit{b}}_{22}& {\mathit{a}}_{21}{\mathit{b}}_{13}+{\mathit{a}}_{22}{\mathit{b}}_{23}\end{array}\right)\text{.}$$

Die Koeffizienten ${\mathit{c}}_{\mathit{i}\mathit{j}}$ entstehen nach der allgemeinen Vorschrift:

$${\mathit{c}}_{\mathit{i}\mathit{j}}=\sum _{\mathit{k}=1}^2{\mathit{a}}_{\mathit{i}\mathit{k}}{\mathit{b}}_{\mathit{k}\mathit{j}}\mathit{i}=1,2\text{und}\mathit{j}=1,2,3.$$

Dies führt zu einer natürlichen Definition der Matrix-Multiplikation. Da $\mathit{A}\mathit{B}\mathit{x}=\mathit{d}=\mathit{C}\mathit{x}$ ist, bezeichnen wir $\mathit{C}=\mathit{A}\mathit{B}$ als Produkt der beiden Matrizen $\mathit{A}$ und $\mathit{B}$

$$\mathit{C}=\mathit{A}\mathit{B}=\left(\begin{array}{rr}{\mathit{a}}_{11}& {\mathit{a}}_{12}\\ {\mathit{a}}_{21}& {\mathit{a}}_{22}\end{array}\right)\left(\begin{array}{ccc}{\mathit{b}}_{11}& {\mathit{b}}_{12}& {\mathit{b}}_{13}\\ {\mathit{b}}_{21}& {\mathit{b}}_{22}& {\mathit{b}}_{23}\end{array}\right).$$