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Mathematics-Online course: Linear Algebra - Matrices - Matrix Operations

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$.


Connection with the composition of linear maps

For the composition of the maps defined by

$\displaystyle w_i = \sum_{k=1}^m a_{ik} v_k,\quad v_k = \sum_{j=1}^s b_{kj} u_j $

we obtain

$\displaystyle w_i = \sum_k \sum_j a_{ik} b_{kj} u_j = \sum_j \left[ \sum_k a_{ik} b_{kj} \right] u_j\, . $

The expression in square brackets is the matrix product and, thus, gives the matrix representation $ C$ of the composition of the two maps.

(Authors: Burkhardt/Höllig/Hörner)

$\displaystyle \left(\begin{array}{rrr} 1 & 10 & 100 \\ 100 & 10 & 1 \end{ar... ...{array}{rrr} 321 & 213 & 132 \\ 123 & 312 & 231 \\ \end{array} \right) $

$\displaystyle \left(\begin{array}{r} 1\\ 2\\ 3 \end{array} \right) \left(\... ...ft(\begin{array}{rr} 7 & 13 \\ 14 & 26 \\ 21 & 39 \end{array} \right) $

$\displaystyle \left(\begin{array}{rr} 3 & 4 \end{array} \right) \left(\begin{array}{r} 3\\ 4 \end{array} \right) = 25 $

$\displaystyle \left(\begin{array}{rr} 1 & \mathrm{i} \\ \mathrm{i} & 1 \en... ... \right) = \left(\begin{array}{r} 0 \\ 2\mathrm{i} \end{array} \right) $

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005