Outer product

Given two vectors

$$ \vec{u}=\sum_i u_i \vec{e}_i,\qquad \vec{v}=\sum_j v_j \vec{e}_j, $$

Their outer product

$$ \vec{u} \otimes \vec{v} = (a_{ij}=u_i v_j) $$

is a matrice of dimension $\dim(\vec u) \times \dim(\vec v)$

Example

For $\vec{u},\vec{v}\in\mathbb{R}^3$

$$ \vec{u} \otimes \vec{v} = \begin{pmatrix} u_1 v_1 & u_1 v_2 & u_1 v_3 \\ u_2 v_1 & u_2 v_2 & u_2 v_3 \\ u_3 v_1 & u_3 v_2 & u_3 v_3 \end{pmatrix}. $$