Vectors Outer Product

Mental Model

The outer product is a mathematical operation that a column vector performs on a row vector.

Given that $a$ is the column vector and $b$ is the row vector,
I like to think of their outer product ($a\otimes b$) as $a$ storing scaled copies of itself by each element in $b$ inside a columns matrix.

$$\left[\begin{array}{c}\uparrow \\ a\\ \downarrow \end{array}\right]\otimes \left[\begin{array}{cccc}{b}_0& b_1& \dots & b_n\end{array}\right]=\left[\begin{array}{cccc}\uparrow & \uparrow & & \uparrow \\ b_0\ast a& b_1\ast a& \dots & b_n\ast a\\ \downarrow & \downarrow & & \downarrow \end{array}\right]$$

Or as $b$ storing scaled copies of itself by each element in $a$ inside a rows matrix.

$$\left[\begin{array}{c}{a}_0\\ a_1\\ ⋮\\ a_m\end{array}\right]\otimes \left[\begin{array}{ccc}\leftarrow & b& \to \end{array}\right]=\left[\begin{array}{ccc}\leftarrow & a_0\ast b& \to \\ \leftarrow & a_1\ast b& \to \\ & ⋮& \\ \leftarrow & a_m\ast b& \to \end{array}\right]$$

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Connections