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arrow_back Can you do matrix multiplication using dot products?

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Can you do matrix multiplication using dot products?

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Dot products provide another way of thinking about matrix multiplication. Recall that if $A=\left[a_{i j}\right]$ is an $m \times r$ matrix and $B=\left[b_{i j}\right]$ is an $r \times n$ matrix, then the $i j$ th entry of $A B$ is $$ a_{i 1} b_{1 j}+a_{i 2} b_{2 j}+\cdots+a_{i r} b_{r j} $$ which is the dot product of the $i$ th row vector of $A$ $$ \left[\begin{array}{llll} a_{i 1} & a_{i 2} & \cdots & a_{i r} \end{array}\right] $$ and the $j$ th column vector of $B$ $$ \left[\begin{array}{c} b_{1 j} \\ b_{2 j} \\ \vdots \\ b_{r j} \end{array}\right] $$ Thus, if the row vectors of $A$ are $\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots, \mathbf{r}_{m}$ and the column vectors of $B$ are $\mathbf{c}_{1}$, $\mathbf{c}_{2}, \ldots, \mathbf{c}_{n}$, then the matrix product $A B$ can be expressed as $$ A B=\left[\begin{array}{cccc} \mathbf{r}_{1} \cdot \mathbf{c}_{1} & \mathbf{r}_{1} \cdot \mathbf{c}_{2} & \cdots & \mathbf{r}_{1} \cdot \mathbf{c}_{n} \\ \mathbf{r}_{2} \cdot \mathbf{c}_{1} & \mathbf{r}_{2} \cdot \mathbf{c}_{2} & \cdots & \mathbf{r}_{2} \cdot \mathbf{c}_{n} \\ \vdots & \vdots & & \vdots \\ \mathbf{r}_{m} \cdot \mathbf{c}_{1} & \mathbf{r}_{m} \cdot \mathbf{c}_{2} & \cdots & \mathbf{r}_{m} \cdot \mathbf{c}_{n} \end{array}\right] $$
by Platinum
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