11.5 Matrix-Matrix Multiplication via Outer Products

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: N/A

Matrix-Matrix Multiplication via Outer Products

let matrices $a \in r^{m \times p}$ and $x \in r^{p \times n}$ . the product $b = a \cdot x$ is the $m \times n$ matrix whose value is given by

B = A (:, 1) X (1, :) + A (:, 2) X (2, :) + \dots + A (:, p) X (p, :)

& notice that this version of matrix-matrix multiplication relies on a sequence of 9.1 outer product of vectors.

Remark. Notice that there are exactly $p$ summands that form the output matrix $B$ .

B = \sum_{k = 1}^{p} A (:, k) X (k, :)

The $k$ th summand is an $m \times n$ matrix which results from taking the outer product between the $k$ th column of $A$ and the $k$ th row of $X$ .
We form the output matrix by summing a sequence of 9.7 Rank-one Update matrices

\begin{aligned} B & = A (:, 1) X (1, :) + A (:, 2) X (2, :) + \dots + A (:, p) \\ = [\begin{array}{c} a_{11} \\ a_{21} \\ ⋮ \\ a_{1 m} \end{array}] [\begin{array}{c} x_{11} & x_{12} & \dots & x_{1 n} \end{array}] + [\begin{array}{c} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{array}] [\begin{array}{c} x_{21} & x_{22} & \dots & x_{2 n} \end{array}] + \dots + [\begin{array}{c} a_{1 p} \\ a_{2 p} \\ ⋮ \\ a_{m p} \end{array}] [\begin{array}{c} x_{p 1} & x_{p 2} & \dots & x_{p n} \end{array}] \end{aligned}

$⧫$