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Let $\pmb{a} = \left[a_1\;a_2\;\dots\;a_n\right], \pmb{b} = \left[b_1\;b_2\;\dots\;b_m\right]$. Then $$K = \left( \begin{array}{ccc} a_1b_1 & a_1b_2 & \cdots & a_1b_m \\ a_2b_1 & a_2b_2 & \cdots & a_2b_m\\ \vdots & \vdots & \ddots & \vdots \\ a_nb_1 & a_nb_2& \cdots& a_nb_m\end{array} \right)$$ is a matrix formed by some sort of product. Is there some way, $\otimes$, to relate these two vectors to $K$? In other words does there exist some operation such that $\pmb{a} \oplus \pmb{b} = K$, or maybe under arbitrary transpositions of $\pmb{a}$ or $\pmb{b}$ there exists a $\otimes$ such that $K$ is the result?

Thanks!

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    $\begingroup$ $\mathbf{a}^\top\mathbf{b}$ is what you want. $\endgroup$ Jul 11, 2015 at 23:00

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This is the outer product $a b^{T}$ for a,b as column vectors.

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