Table of Contents

Start with

$$ \begin{aligned} A & = \begin{bmatrix} a_{11} & a_{12} & * & a_{1n} \\ a_{21} & a_{22} & * & a_{2n} \\ * & * & * & * \\ a_{m1} & a_{m2} & * & a_{mn} \end{bmatrix} \\ B & = \begin{bmatrix} b_{11} & b_{12} & * & b_{1p} \\ b_{21} & b_{22} & * & b_{2p} \\ * & * & * & * \\ b_{n1} & b_{n2} & * & b_{np} \end{bmatrix} \end{aligned} $$

The usual way (or the definition):

$$ (AB)_{ij} = (\text{row} \ i \ \text{of} \ A) \sdot (\text{column} \ j \ \text{of} \ B) $$

As a linear combination of the columns of A (recommended):

$$ AB = A\begin{bmatrix} \boldsymbol{b_1} & \boldsymbol{b_2} & * & \boldsymbol{b_p} \end{bmatrix} = \begin{bmatrix} A\boldsymbol{b_1} & A\boldsymbol{b_2} & * & A\boldsymbol{b_p} \end{bmatrix} $$

$$ A\boldsymbol{b_i} = b_{1i}\begin{bmatrix} a_{11} \\ a_{21} \\ * \\ a_{m1} \end{bmatrix} + b_{2i}\begin{bmatrix} a_{12} \\ a_{22} \\ * \\ a_{m2} \end{bmatrix} + … + b_{ni}\begin{bmatrix} a_{1n} \\ a_{2n} \\ * \\ a_{mn} \end{bmatrix} $$

As a linear combination of the rows of B:

$$ AB = \begin{bmatrix} \boldsymbol{a_1} \\ \boldsymbol{a_2} \\ * \\ \boldsymbol{a_m} \end{bmatrix} B = \begin{bmatrix} \boldsymbol{a_1}B \\ \boldsymbol{a_2}B \\ * \\ \boldsymbol{a_m}B \end{bmatrix} $$

$$ \begin{aligned} \boldsymbol{a_i} B = & a_{i1} \begin{bmatrix} b_{11} & b_{12} & * & b_{1p} \end{bmatrix} & + \\ & a_{i2}\begin{bmatrix} b_{21} & b_{22} & * & b_{2p} \end{bmatrix} & + \\ & … & + \\ & a_{in}\begin{bmatrix} b_{n1} & b_{n2} & * & b_{np} \end{bmatrix} \end{aligned} $$

As the sum of rank one matrices (important):

$$ \begin{aligned} AB & = \sum_{i=0}^n (\text{column} \ i \ \text{of} \ A) (\text{row} \ i \ \text{of} \ B) \\ & = \sum_{i=0}^n \begin{bmatrix} a_{1i} \\ a_{2i} \\ * \\ a_{mi} \end{bmatrix} \begin{bmatrix} b_{i1} & b_{i2} & * & b_{ip} \end{bmatrix} \end{aligned} $$