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Matrix Multiplication

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The product of two matrices $ A \in K^{n \times m}$ and $ B \in K^{m \times s}$ is the $ (n \times s)$-Matrix $ AB := (c_{ij})$ defined by

$\displaystyle c_{ij} = \sum_{k=1}^m a_{ik}b_{kj}\, . $

The product is defined only if the number of columns of $ A$ equals the number of rows of $ B$.

We can interpret the matrix product as follows: The element $ c_{ij}$ of product matrix $ (c_{ij}) = AB$ is the ,,scalar product`` of the $ i$-th row vector of $ A$ and the $ j$-th column vector of $ B$:

$\displaystyle \left( \begin{array}{ccccc} & & \vphantom{b_{1j}} & & \\ & & ... ...& \vphantom{\vdots} & & \\ & & \vphantom{b_{mj}} & & \end{array} \right) $

Matrix multiplication corresponds to composition of linear maps

$\displaystyle \alpha: u \mapsto v=Bu,\quad \beta: v \mapsto w=Av\, , $

that is, $ C$ is the matrix representation of $ \beta\circ\alpha$.


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  automatically generated 5/23/2011