Definition:Outer Product

Definition
Given two vectors $$U=(u_1,u_2,...,u_m)$$ and $$V=(v_1,v_2,...,v_n)$$, their outer product $$U\otimes V$$ is defined as

$$U\otimes V=A=\begin{bmatrix}u_1v_1 & u_1v_2 & \dots & u_1v_n \\ u_2v_1 & u_2v_2 & \dots & u_2v_n \\ \vdots & \vdots & \ddots & \vdots\\ u_mv_1 & u_mv_2 & \dots & u_mv_n \end{bmatrix}$$

Index Notation
Given two vectors $$u_i$$ and $$v_j$$, their outer product $$u_i\otimes v_j$$ is defined as

$$u_i \otimes v_j = a_{ij}=u_iv_j$$

Matrix Multiplication
Given two column vectors $$\mathbf{u}$$ and $$\mathbf{v}$$, their outer product $$\mathbf{u}\otimes\mathbf{v}$$ is defined as

$$\mathbf{u}\otimes\mathbf{v}=A=\mathbf{u}\mathbf{v}^T$$

Properties
$$A\mathbf{v}=\mathbf{u}||\mathbf{v}||^2$$     See Vector Length.

$$\mathbf{u}\otimes\mathbf{v}=(\mathbf{v}\otimes\mathbf{u})^T$$