Definition:Outer Product

Definition
Given two vectors $$\vec U = \left({u_1, u_2, \ldots, u_m}\right)$$ and $$\vec V = \left({v_1, v_2, \ldots ,v_n}\right)$$, their outer product $$\vec U \otimes \vec V$$ is defined as:


 * $$U \otimes V = A = \begin{bmatrix}

u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\ u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\ \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end{bmatrix}$$

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


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

Matrix Multiplication
Given two vectors expressed as column matrices $$\mathbf{u}$$ 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

 * $$A \mathbf{v} = \mathbf{u} \left\|{\mathbf{v}}\right\|^2$$: see Vector Length.


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